Critter Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠤⣤⣤⣤⣤⣄⣀⣉⣓⡒⠒⠒⠒⠒⠒⠒⠂⠐⠀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣉⠉⠉⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣧⡀⣾⣿⣿⣿⣿⣿⡄⣠⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⠛⠻⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠸⠷⠿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⠿⠿⢆⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⢿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠲⡟⡊⣼⣔⠆⠟⡾⣀⠶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠛⢷⣦⢅⠰⠰⠀⠂⠈⣿⣷⠖⠀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢁⣟⣳⣤⡛⠈⡀⢤⣤⣴⡶⢿⠉⠀⠀⠀⠉⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⠏⠁⠀⠀⠀⡻⣿⣿⣾⠃⠀⠀⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡠⠙⠋⢀⣾⣧⡀⠀⠀⠈⠑⠋⠀⠀⢠⡇⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⠟⠘⠒⠋⠛⠙⠶⢄⠀⠀⠀⠀⠀⠀⠊⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣿⡿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠏⠁⠌⠀⠀⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⠏⣯⠐⠄⠦⢣⣠⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣻⣉⠟⣔⡼⣆⡓⠀⣤⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢟⣿⡟⢽⠞⠏⢈⡹⢘⡸⡟⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠞⡜⢳⠒⣥⠬⢱⣯⡂⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣬⣡⢸⡧⢞⣷⣿⣻⣿⡓⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣭⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣷⡶⢔⣧⣴⡿⣾⠿⠱⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣿⣿⡏⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⠟⣷⠂⣿⣾⡟⠋⣽⣯⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣻⣿⣿⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⢹⣷⢻⡷⠶⠀⠰⠰⢿⠷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣷⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢻⡧⢞⣋⡙⠟⠧⢭⡏⠩⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣽⡿⣓⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⡾⢿⣿⣔⣿⣟⠃⢳⠷⠂⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠛⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣼⣾⣟⡟⡒⢬⡿⣼⢚⡟⠂⣴⠀⢂⣤⣶⣾⣿⣿⡿⠏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⡹⠞⣿⣼⣿⣿⡱⣜⣿⣶⣼⣿⣿⣿⣿⣷⣤⣤⣴⣿⣅⡀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⡷⢆⡻⣯⣾⢋⣿⣷⣿⣿⣿⣿⣿⡿⣿⣿⠿⣟⡋⠿⠉⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣎⡷⣽⡿⣻⡦⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠉⠁⠈⠀⠉⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣳⠟⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣗⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡼⣿⣦⣀⣀⠀⠀⠀⢀⣀⣀⣀⣠⣀⠀⣾⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⢹⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⢿⣿⣿⣿⣿⡿⡿⣿⢹⡿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣀⠹⢿⣿⣿⠧⣷⣽⡾⢿⣿⣿⣿⡟⣺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣿⣿⡄⢈⡷⢿⠀⢘⣿⡡⣾⣻⣿⣿⡄⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠀⢹⣿⣿⣧⡎⠛⠈⠀⠀⠉⠷⢹⣿⣿⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣿⣿⣋⡀⠀⠈⠛⠛⣿⡁⠀⠀⠃⠐⣀⣼⣿⣿⣾⣼⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢹⣿⣾⣷⡘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠂⠀⠉⠙⠛⠋⠉⡡⠄⠀⠺⣿⡟⠋⠀⠀⠙⣿⣟⡃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⡶⠶⠿⠿⢿⣿⣿⣿⣿⡏⠉⠀⠉⠻⣝⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣴⣿⣿⣿⣿⣿⠇⢠⠉⠀⠈⡙⡻⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣄⡐⢠⣀⣤⣨⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⡿⠿⠻⠟⠛⠛⠛⠈⠛⠉⠉⠉⠛⠛⠛⠛⠛⠿⣿⡿⡿⠿⣿⡿⠿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⢷⠀⠒⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣤⣤⣀⠀⠀⢀⡾⠃⡔⠀⠘⠆⠊⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣷⡀⣾⠁⠀⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣩⣤⣤⣶⣶⣤⣤⣀⡈⠉⠓⠏⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠃⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⣽⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⢀⣤⣶⠟⠋⠉⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⢀⣀⣤⣴⣾⣿⣿⠀⠒⢶⣶⣾⡏⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠘⢡⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠠⠞⠛⠋⠉⠉⠉⠉⠙⠓⠢⠤⣈⡙⠦⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⢀⣴⠂⠀⠀⠀⢀⡇⠀⣾⣿⠁⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠈⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠁⣠⣿⠏⠀⠀⠀⠀⢸⡇⠀⢻⣿⣀⣾⠋⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⣄⡀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠸⢁⣼⣿⡟⠀⠀⠀⠀⠀⣿⣇⠀⠘⣿⣿⣿⢰⡟⠁⢀⠀⠀⠀⠀⢀⣤⠀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⠀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⡄⠀⠈⢿⣿⣿⣧⢠⡟⠀⠀⠀⣰⣿⣁⣈⣉⡉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠙⣿⠏
⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⠀⠀⣰⠀⠀⠀⢸⣿⣷⡄⠀⠀⠉⠛⢿⣿⣷⡀⠀⢰⣿⡿⠛⣩⡥⠤⠤⢤⣄⣀⠀⠁⠒⠐⡒⠚⠉
⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⣰⣿⠀⠀⠀⠀⢿⣿⣿⣦⠀⠀⠀⠀⠀⠉⠉⠀⠸⣿⢁⣾⠋⠀⠀⠀⠀⠀⠙⠻⠷⠾⠋⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢠⣿⣿⠀⠀⢀⡀⠈⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠻⠘⣿
⠀⠀⠀⠀⠀⢸⡏⢸⣿⠀⣼⣿⣿⠀⠀⢸⣇⠀⠈⠻⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠸⠃⢸⣿⠀⣿⣿⣿⡇⠀⠘⣿⡀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⣿⣿⣿⣿⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣧⡀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹⣿⣿⣿⣧⠀⠀⠙⢿⣶⣄⠀⡀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣄⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣿⣿⣿⣿⣧⠀⠀⠀⠙⢿⣷⣜⣦⡀⠀⠀⣰⠀⠀⠈⢻⣿⡆⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⢻⣿⣧⡀⠀⠀⠀⠙⣿⣿⣿⡄⠀⢹⣇⠀⠀⠀⠹⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠹⣿⣷⣄⠀⠀⠀⠈⢿⣿⣿⡀⠀⢻⣧⣀⣧⠀⢹⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠈⠻⣿⣦⡀⠀⠀⠈⠃⣿⡇⠀⠀⢻⣿⣿⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀⠀⠀⡿⠀⠀⠀⢸⣿⠇⠀⠀⠀⠈⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⣠⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠤⣤⣤⣤⣤⣄⣀⣉⣓⡒⠒⠒⠒⠒⠒⠒⠂⠐⠀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣉⠉⠉⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣧⡀⣾⣿⣿⣿⣿⣿⡄⣠⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⠛⠻⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠸⠷⠿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⠿⠿⢆⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⢿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋

⠀⠀⠀⠱⠀⠀⢂⡀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⣤⣤⣾⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣅⣡⣄⣻⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⢿⣿⠻⡧⠈
⠀⠀⠀⠀⠀⠀⣠⣴⣿⡴⢻⠤⣤⣤⠀⠀⠀⠀⠙⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢴⡿⡏⠹⠀⢨⠈⠟⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢄⠀⠀⠈⠉
⣶⠀⠀⠀⢸⣟⡇⠀⠀⠀⢸⠀⠀⢹⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⢀
⣽⡄⠀⠀⢈⣭⠽⠀⠀⣶⣾⠀⠀⠲⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠄⠀⠀⠀⠀⠀⣞⣥⣀⠀⢀
⣿⡇⠀⠀⠀⠀⢡⠄⠀⠀⠀⠀⣠⣲⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠈⣟⣿⣿⠣⠸
⡻⠁⠀⠀⠀⠐⠫⢹⢿⣿⣏⣶⣾⠏⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣿⡷⠰⣿⣿⡗
⠀⠀⠀⡀⠀⠀⠀⣀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠁⠀⠹⡿⠁⠀⠀⠀⠀⠀⠀⠐⣄⠀⡦⠶⢢⡀⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠁⠀⠹⣴⣦⣄
⠀⠀⠀⠀⠂⡴⡦⠐⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠀⣂⡏⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣢⣤⣤⣴⣇⣛⣛
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡀⠤⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⢲⠒
⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠜⢷⠃⠀⠀⡀⠀⠀⠹⣮⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠂⢠⣶⣄⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⠘⠛⠛⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣠⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠚⠓⣶⣶⠂⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠉⠉⠉⢹⣿⣿⡿
⠀⠀⠀⠀⠀⠻⣿⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣾⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡏⠀⠀⠀⢸⣿⣿⡶
⠀⠀⠀⡖⠂⣦⣶⡆⢚⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠊⠃⠉⠖⠚⠃⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⢶⠆⢸⣿⣿⣶⠄

⠀⠀⠀⠀⠀⠀⠀⣄
⠀⢿⢿⣦⡇⠀⠀⠀⠀⢠⣶⣾⣷⢄⣀⠀⠀⠀⠄⠀⢀⡀⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣏⠀⠀⠀⠠⠦⡐⢿⣾⠋⠀⠈⠙⠻⡀⠀⠀⠀⠀⠐⠖⠚⠻⠟⠛⠇⠀⠀⠀⠈⠀⠀⠀⠰⠶⠶⠶⠦⠑⠺⠾⣷⣮⡂⡀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⢸⣧⣶⣦⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣗
⠀⠀⢦⣈⡤⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣇⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⣄⠀⠀⡆⠘⣿
⠀⠀⠙⠿⠿⠶⠿⣿⣿⣿⡞⢹⣿⣿⣿⣿⡏⠀⠀⠀⠐⣦⣀⣄⡀⠀⠀⠄⢀⡀⠀⢠⠀⠀⠀⠀⠀⣼⣿⡾⡣⠊⢿⣷⣿⣇⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣇⣀⠙⣿⣿⣿⢆⣴⣾⡆⠀⠈⢿⣿⣷⣤⣸⣾⣼⣧⣤⣿⣇⣾⣤⢠⣾⣿⣿⣿⣜⢄⡀⠊⠀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠙⠛⢿⣻⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣦⣂⣤⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠠⢸⣿⣿⣿⣧⡈⣻⠟⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣃⣀⣈⣛⢿⣿⣿⣿⣿⣿⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣄⣄⢿⣿⣸⣿⣿⣏⣀⣜⣿⣿⣿⣿⣿⣿⣿⣿⡟⡉⢘⡩⠙⠉⢿⣭⣏⣹⡾⣿⡟⠛⢿⣮⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⢭⣶⣿⣿⣿⢧⠉⠁⠀⠈⠉⠉⠛⠛⠻⣿⣿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⠀⠈⢳⣤⣿⡿⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣮⠀⠀⠹⣿⣇⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣡⠀⠀⠀⠀⢹⣿⡷⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠛⠁⠀⠀⠀⠀⠀⣟⠣⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣯⡝⢿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⡡⠀⠀⠀⠀⠀⠀⠀⠓⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⠷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠰⡟⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠼⠓⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠷⣹⣸⣿⣿⡟⡿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⢀⣠⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠃⡀⢦⣦⡀⣠⣴⠞⣻⣿⠿⠛⢿⡇⠘⠘⠋⠋⠁⠹⣟⠿⠻⠟⠒⠈⠉⢉⡈⣩⡟
⠀⠀⠀⠀⠀⠀⠀⢶⣿⣾⡅⣀⢠⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠿⠀⠀⠀⠈⠁⠻⢀⠹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣯⡴⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⡇⢸⡄⠀⠠⣇⠀⡀⢀⠀⠀⠀⠀⠀⠀⠾⠅
⠀⠀⠀⠀⠀⠀⠀⢿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⠇⠀⢼⣿⢲⠇⢰⡄⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣷⣶⠟⠁⠀⣆⠀⡀⠀⠀⠁⠀⠀⠀⠀⠛⠀⠀⠘⠃⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠈⠀⠀⠠⠷⣄⠀⠀⠀⠀⠀⠀⠀⣴⠀⠖⠳⠂⢀⠒⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠀⢀⡈⠲⠶⢤⠲⣶⣤⣿⡀⠀⠀⠀⠀⢀⡞⢁⣀⣤⣤⣦⡠⢾⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣦⡄⠉⢂⣀⣀⠀⠀⡸⠿⣃⡄⠀⠀⠀⠸⣷⡌⡘⢻⣿⠦⢵⡎
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣴⢶⣟⠉⠁⠀⠀⠀⠈⠻⣿⣄⠀⢀⣿⠀⠀⠀⠀⣙⡛⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣯⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠙⢧⣠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⡺⠛⠛⠿⠿⢧⣀⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣽⡤⠆⠙⠆⠀⠀⠀⠀⣻⣣⣿⣦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢶⠀⠀⠀⠀⠐⢾⣿⡿⢫⠙⠃⠀⢤⣤⣶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣄⢰⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣴⣦⣤⣤⣤⣶⣿⣿⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣭⣇⣿⣿⣿⢋⡿⡝⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⡀⠀⣀⣤⣄⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢢⣱⣀⢀⣠⣤⡞⣻⣿⡿⠟⠻⣿⡄⠘⠘⠛⠸⠉⠈⢹⡛⡿⠻⠿⠳⠀⠉⠉⢉⠈⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⣏⠉⠈⣉⣤⠶⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⡀⠠⠘⡷⠀⠀⠀⠀⢪⠑⣆⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⠞⠁⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⣄⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⡕⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡀⢸⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⠸⡇⠀⠀⣿⣤⢸⠀⢐⠀⠀⠀⠀⠀⠀⠀⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⣧⣄⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠈⠇⠀⢁⠀⠘⢿⡇⠛⠀⢸⡇⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⠟⠃⠀⠀⢿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠃⠀⠀⠀⠀⠀⠁⠀⠀⠚⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠛⠩⠛⠁⡠⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⠀⠀⣀⠈⠲⠾⢤⠐⢶⣦⣴⣿⠀⠀⠀⠀⠀⠀⢠⡏⠀⣀⣤⣤⣴⣄⡤⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣷⣄⣈⠛⠂⠀⢄⠀⠀⠀⣿⠿⢇⡀⠀⠀⠀⠀⢼⣧⣈⠩⢿⣿⣦⠬⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⣤⣄⣿⡛⠋⠀⠀⠀⠀⠒⢿⣿⡀⠀⠀⠐⡆⠉⠀⠁⠈⠛⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣿⡿⠯⠟⣉⣉⡓⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⠁⢰⡟⠉⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⢻⣾⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠟⢯⡉⠀⠉⠉⠛⠻⣶⡶⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⣾⣶⠊⠀⠉⠀⠀⡀⠀⠀⣰⣧⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠤⠀⠀⠀⠀⠈⢛⣿⣿⠋⡄⠉⠀⠠⠴⠶⠾⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣳⢄⠀⠀⡀⢠⡄⣠⣀⠀⣠⣶⢫⣿⡻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⠘⠊⠛⠦⠙⠈⣋⢿⠉⠉⠙⠟⣸⣿⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣁⣠⠿⠗⠀⠂⣡⠞⣈⢀⠀⠈⠻⣿⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⣠⡀⣠⣼⠂⠀⠀⠘⠟⡄⠀⣀⢰⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡿⢾⠟⠀⠀⡄⠀⠀⠀⠱⣾⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢼⠧⠠⡀⠸⣆⣤⡠⠀⠀⠀⠀⣬⡆⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢤⡶⠀⣸⣴⠟⠈⠋⠛⠈⠀⣀⠀⠘⡇⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢾⡏⠀⢀⡉⠻⢧⣤⣤⡾⠟⣉⡃⠀⣷⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣸⠃⢐⠄⢂⣀⣀⣼⡁⡀⡆⠨⡇⠀⢹⡗⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡯⡆⠀⠋⢋⣉⡤⠾⢧⣍⣠⡼⡆⢀⡿⠳⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⢷⢧⣤⣀⣈⠀⢀⡀⠉⠛⡁⠁⢿⠃⡀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⢾⣟⣬⣾⣿⣿⣿⡿⣷⢦⠳⠏⢠⣈⡀⠉⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢹⣿⣿⣿⣿⣿⣿⠷⢹⡘⢾⣶⣤⣭⢙⡀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⢽⣿⣿⣿⣿⣟⡿⣎⠳⣶⣿⡺⠿⢦⠍⠀⠀⠰⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠉⢁⠘⣿⢿⣿⣿⣟⣼⣼⠿⢾⣍⠇⠀⡄⠀⠀⠑⠴⠣⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢿⡇⠀⣤⣶⣻⢿⣼⢿⡠⣿⡆⠠⡤⠄⠰⣂⠀⣤⡀⣆⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⡀⢄⣠⠄⠉⠻⢋⡿⠾⠏⡍⠈⡀⠀⠀⠀⠀⡍⠗⠀⡻⣇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⢈⡀⠱⠛⠊⣴⠄⠀⠐⡄⠄⢀⠀⠘⠀⠑⢚⠀⠀⣶⠌⢻⡆⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣺⠗⠂⠀⡡⢿⡆⣶⣠⠁⠀⢸⡆⠠⣄⠈⠁⠤⠀⠈⠳⢦⣷⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣲⣖⠠⠞⣡⣾⣷⣿⣇⣰⣶⠎⣷⣵⡈⠠⠦⡀⠈⠀⠀⠈⣂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠶⠿⠿⢳⣻⣿⣿⣹⡿⡴⡄⡸⠃⠰⡖⢷⡼⣒⣶⠾⠿⠿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣻⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢶⠀⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⢳⡄⢀⠁⡀⠀⠀⠀⠠⠀⠀⠀⠠⠤⣾⣿⣷⡇⠐⠑
⠀⠀⠀⠐⣦⠀⠈⣀⠏⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⣟⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⡆⠀⠋⢠⡿⠀⠀⠀⠀⠀⠐⠾⢦⠄⠀⢀⡄⢿⣶⣦⠎⠀⠀⠀⠀⠀⠂⠸⠀⢼
⠀⠀⠀⢀⢀⠛⣦⡸⣏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠄
⠀⠀⠀⠈⣞⢁⠛⣇⢙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠁⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢸⣶⡟⠀⠀⠀⠀⠀⠠⠄⠐⠶⠲⠶⠂⠒⠀⠀⠀⠐⠊⠋⠉⠉⠉⠀⢰⠔⠊
⠀⠀⠀⠀⢠⣀⠀⠀⣸⣷⡌⢛⠀⠀⠀⠀⠀⣀⠀⠀⡄⠀⢀⡀⢀⡄⠀⠀⢠⡆⢠⣇⠀⠀⠄
⠀⠀⠠⠀⡼⠈⠁⠀⠀⠀⢡⢘⡜⠀⠀⣰⡷⠋⠂⠀⠀⠀⠈⠀⠛⠀⠀⠀⠘⣡⠃⠀⠀⠀⠀⢰
⠀⠀⢢⡎⢵⣠⠄⠀⠀⠀⠈⠿⠇⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠛⠄
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣀⡄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠢⣹⣿⡷⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠐⠛⠑⠂⣍⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠎⢐⣴⢰⣿⣿⣶⣼⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡏⣿⣼⡎⣫⣿⠻⣎⣿⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⡹⢷⣿⢫⣶⠈⢻⣿⢶⣝⠻⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣇⢹⡀⠛⢸⣯⢶⡘⣿⣠⡎⠓⢧⣽⣿⣟⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⣇⣌⢦⣘⣿⡜⣗⣌⠳⣝⢦⢠⣿⣿⣟⢿⣿⣯⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡶⣼⣗⠿⣟⡿⢾⣿⣶⣌⠻⢷⣿⡿⣿⢧⡙⣯⡻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣿⠹⣦⢸⡆⣆⠈⠛⠿⣿⢶⣿⣿⡜⢧⠹⣌⢻⡌⢿⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⣙⠾⣽⣶⣦⣤⣀⡀⠈⠙⠎⠳⣜⣮⣷⡀⢍⡈⠙⣿⠻⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⣽⡲⠦⣭⣙⠻⢿⣷⣬⠑⢶⡶⣭⣍⣛⠺⣿⣷⣮⣿⣶⣮⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣬⣙⠳⢭⣻⣷⡌⢿⢹⣙⣛⣿⣶⣭⣿⣿⣿⣿⣿⣿⣬⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠛⣦⡹⣿⡇⠘⢸⡇⠈⠉⠉⠉⠛⠛⠛⠿⠿⣿⣏⣷⣶⣬⣔⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣼⡟⠐⡸⡇⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣿⡿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣤⡤⠤⠾⠁⢿⣀⠀⣳⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠚⠻⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠙⢀⣚⣛⣛⡛⣀⡿⣐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢛⣉⠭⠭⠭⠭⠤⠭⠿⠂⠡⠀⠦

⠀⠀⠀⠀⠀⠀⢀⣤⠀⣺⠳⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠠⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢂⡀⢀⡼⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⡄⠀⡄⠀⠀⠀⠀⠠⠂⠀⠀⠀⠒⠒⠒⣿⣿⣷⣿⠂⠈⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢧⡀⠀⠁⠀⡿⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⣿⡃⠀⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠘⣧⠀⠠⡟⠁⣰⣿⠀⠀⠀⠀⠀⠀⠀⢤⣶⢤⡀⠀⠀⠀⡀⢹⣷⣴⣀⡴⠋⠀⠀⠀⠀⠀⠰⠀⠀⡆⢀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⣄⠀⢠⣴⠏⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠉⠉⠀⠐⠃⠈⣿⣿⡟⠁⠀⠀⠀⠀⠁⠀⠀⢀⠐⠁⠀⠹
⠀⠀⠀⠀⢰⢀⣆⢸⣿⡄⢳⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⣿⠁⡘⠉⣧⠈⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠉⠀⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠤⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠋⢹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣤⡤⣤⣴⠶⠚⠟⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠸⣿⡟⠁⠀⠀⠀⠀⠀⠀⠐⠒⠋⠉⠙⠉⠛⠛⠁⠈⠀⠀⠀⠀⠈⠉⠁⡃⠈⠀⠀⠀⠀⠀⡗⠈
⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⣿⣦⡌⢻⠁⠀⠀⠀⠀⠀⠄⢀⠀⢀⢰⡀⠀⠀⢀⠀⠀⡀⠀⠀⠀⢀⡆⠀⢰⣇⠀⠀⢀⡄
⠀⠀⠀⢠⠀⢠⡏⠹⠀⠉⠉⠁⠀⢇⠀⣇⡆⠀⠀⠀⣼⣴⠏⡇⠀⠈⠁⠀⠉⠉⠀⢻⠁⠀⠀⠀⠸⠇⣰⠃⠈⠁⠀⠀⠀⠠
⠀⠀⠀⡇⣠⢿⠀⡀⠀⠀⠀⠀⠀⠸⣦⣿⠁⠀⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⣿⣇⠘⠓⠓⠂⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠸⠀⠀⠘⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣷⣦⣤⣤⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡄⣣⡍⢿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⡿⣿⡿⢿⡇⠉
⠀⠀⠀⠀⠀⠀⡀⣄⣮⣧⡼⣧⣀⣤⡀⠀⠀⠀⠀⠻⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⢠⣽⣿⠋⠃⠈⡁⠺⢿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠊⠁
⣤⠀⠀⠀⠀⡶⣶⠈⡀⠂⠀⡇⠀⠀⣯⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⣹⡀⠀⠀⠐⣛⣛⡆⠀⢠⣤⡇⠀⢰⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⢀⠀⠀⠀⠀⠀⢀⡟⣋⠉⠁⠉
⣿⡇⠀⠀⠀⠩⢒⢃⠀⠈⠉⠁⠀⠀⣾⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⡆⠀⠀⠀⠀⠀⢸⡿⣿⣿⢄⢼
⢿⠇⠀⠀⠀⠀⠴⡳⣴⣶⣶⢒⣬⣿⡻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠚⠚⠉⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⣤⣤⣤⢀⣴⣦⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠀⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⠀⢠
⠀⠀⠠⠒⠀⠀⠀⠐⠒⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣀⣀⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠼⠍⠡⡾⣂⡀
⠀⠀⠀⠀⢀⠀⠀⠀⡀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⣸⣿⣿⣖
⠀⠀⠀⠀⠀⠚⠛⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠀⠀⠀⣾⡿⣿⡧
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣿⡏⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⠀⠀⠀⠀⠀⣀⢀⡠⠐⠒⠒⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠏⠛⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠍⢛⠀⠀⠀⠄⠀⠀⢘⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠁⣰⣿⣆⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡉⠛⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⣠⣶⣄⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣀⣄
⠀⠀⠀⠘⠉⢲⣶⣖⠙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡇⠈⠈⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠄⠸⢿⡟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⠃⠀⠀⠀⢿⣿⣿⡶
⠀⠀⠀⡸⠒⢶⣶⡶⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠊⠂⠉⠖⠚⠃⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠰⣶⠆⢸⣿⣿⣶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⢻⣿⣦⣄⣀⠀⠀⢀⣀⣠⣤⣠⣄⡄⣾⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⡟⣿⣿⣿⣿⡿⣿⣿⢹⣿⣿⣿⣿⣿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣥⡀⠻⣿⢿⠛⢻⣾⡿⢛⣿⣿⣿⡏⣹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢻⣿⣿⡗⢀⣾⡘⡆⠀⣽⣞⣾⣼⣿⣿⣧⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠁⠀⣿⣿⣿⡟⠈⠁⢀⢀⠀⢘⡠⣿⢻⠟⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣤⣄⡀⠈⠉⢽⣷⡀⣀⠈⣆⣦⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⡻⠿⢿⣿⣿⣿⡍⠀⠀⢀⣿⣿⣿⠇⠘⠻⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⠉⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠛⠛⠀⠀⡀⣀⣿⣿⣦⣼⣿⣿⣿⡿⠿⠛⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⠉⢉⣹⣿⣿⣿⣿⡇⠀⠄⠀⠩⣮⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⢿⣿⣿⣿⣿⠁⢈⠀⠀⠀⠠⣬⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣽⣶⣷⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣿⣿⣿⣿⡿⣿⣿⢿⠿⠛⠿⠿⠿⠻⢿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠛⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠈⠙⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣀⡄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠢⣹⣿⡷⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠐⠛⠑⠂⣍⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠎⢐⣴⢰⣿⣿⣶⣼⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡏⣿⣼⡎⣫⣿⠻⣎⣿⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⡹⢷⣿⢫⣶⠈⢻⣿⢶⣝⠻⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣇⢹⡀⠛⢸⣯⢶⡘⣿⣠⡎⠓⢧⣽⣿⣟⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⣇⣌⢦⣘⣿⡜⣗⣌⠳⣝⢦⢠⣿⣿⣟⢿⣿⣯⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡶⣼⣗⠿⣟⡿⢾⣿⣶⣌⠻⢷⣿⡿⣿⢧⡙⣯⡻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣿⠹⣦⢸⡆⣆⠈⠛⠿⣿⢶⣿⣿⡜⢧⠹⣌⢻⡌⢿⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⣙⠾⣽⣶⣦⣤⣀⡀⠈⠙⠎⠳⣜⣮⣷⡀⢍⡈⠙⣿⠻⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⣽⡲⠦⣭⣙⠻⢿⣷⣬⠑⢶⡶⣭⣍⣛⠺⣿⣷⣮⣿⣶⣮⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣬⣙⠳⢭⣻⣷⡌⢿⢹⣙⣛⣿⣶⣭⣿⣿⣿⣿⣿⣿⣬⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠛⣦⡹⣿⡇⠘⢸⡇⠈⠉⠉⠉⠛⠛⠛⠿⠿⣿⣏⣷⣶⣬⣔⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣼⡟⠐⡸⡇⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣿⡿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣤⡤⠤⠾⠁⢿⣀⠀⣳⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠚⠻⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠙⢀⣚⣛⣛⡛⣀⡿⣐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢛⣉⠭⠭⠭⠭⠤⠭⠿⠂⠡⠀⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢻⠀⠚⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣶⣶⣶⣦⣄⡀⠀⢀⣼⠋⡴⠀⠈⠇⠊⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠾⠿⠿⠿⣿⣿⣆⢸⠃⠀⢇⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣤⣄⡀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣭⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⢀⣠⣴⠖⠂⠀⠀⢀⣀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠞⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⣀⣤⣶⣾⣿⣿⣿⣄⠈⠛⢿⣿⠇⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠁⣼⡿⠁⢀⠀⠀⠀⠀⢀⠀⢀⣿⡟⠀⠊⠉⠀⠀⠀⠀⠀⠀⠉⠁⠒⠦⣌⣓⠀⠀⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⣼⠟⢀⣴⡏⠀⠀⠀⠀⣾⠀⢸⣿⡇⢀⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⢰⠏⣠⣾⡟⠀⠀⠀⠀⢀⣿⠀⠈⣿⣧⣾⡇⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠦⠄⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣰⣿⣿⠃⠀⠀⠀⠀⢸⣿⡄⠀⠘⣿⣿⣷⣿⠀⣠⠀⠀⠀⠀⣠⡞⠡⢶⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠀⠀⢀⠀⠀⠘⣿⣷⡀⠀⠈⠻⣿⣿⣤⣿⠀⠀⠀⣼⣿⣶⠶⠶⠂⠀⠀⠀⠀⠀⢄⣀⢀⣀⣬⠟
⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⣰⡇⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠉⠛⠻⠧⠀⠠⣿⡟⣡⡾⠛⠉⠉⠙⠓⢶⣤⣀⣠⡥
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢠⣿⡇⠀⠀⠀⠈⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢿⢰⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢸⡟⣿⣿⠀⣿⣿⡇⠀⠀⣶⠀⠈⢿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠈⠀⠻⣆
⠀⠀⠀⠀⠀⢸⠃⢿⣿⢀⣿⣿⣷⠀⠀⢹⡆⠀⠀⠙⠿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠈⠀⢸⣿⠈⣿⣿⣿⡆⠀⠸⣿⡄⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣷⣄⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣿⣿⣿⣷⡀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⡀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢻⣿⣿⣿⣷⡀⠀⠀⠙⢿⣷⡜⣦⡀⠀⠀⢀⡀⠀⠙⢿⣿⡄⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠻⣿⣷⡀⠀⠀⠀⠙⢿⣾⣿⣄⠀⢸⣇⠀⠀⠀⢻⣷⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠘⢿⣿⣄⠀⠀⠀⠈⢻⣿⣿⡄⠀⢻⣦⡀⣦⠀⢻⡀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠈⠻⣿⣷⡀⠀⠀⠀⠋⢻⡇⠀⠀⢻⣿⣿⠀⠈⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⢸⠃⠀⠀⠘⣿⡏⠀⠀⠀⠘⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⢠⠟
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠔⣠⢀⡐⣢⢄⡄⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣮⣦⣘⠱⡏⠛⠳⠃⠓⠺⣵⣮⡋⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢃⣀⠉⢿⡗⢦⠘⠙⠁⠉⠀⣸⣿⣟⠁⠀⠶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢁⣮⣭⣷⣶⠁⠀⠄⠶⡶⣶⣿⡻⣇⠀⠀⠀⠀⠀⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⣿⡿⠉⠀⡀⠂⠀⡹⣿⣿⣷⡏⠀⠀⠀⡄⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡠⠍⠛⠁⢸⣿⣇⡀⠀⠀⠀⠐⠚⠁⠀⠀⢰⠇⠀⠀⢨⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠒⠲⠒⠛⠟⠓⢴⣀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣟⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢿⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⡟⠃⡠⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢿⡟⢀⠀⡁⣈⠠⠀⠢⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣽⣿⢨⣿⢀⢁⡉⣸⠲⠓⢡⣟⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣯⣬⣑⢮⣟⢡⣭⣁⠸⣞⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢛⣭⣿⠃⠾⠛⡃⢨⡔⢫⡅⠻⠙⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣗⢳⠛⣶⠋⣤⠾⠍⣿⣖⢠⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣯⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣍⣤⣽⡥⣟⣷⣿⣛⣿⡟⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣭⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⣹⣖⡤⣐⣇⣨⡿⣶⡿⡟⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⢾⠯⠸⣗⣽⡻⡟⣻⣟⣵⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⣯⡳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠱⣾⡳⢾⣭⣭⠀⢠⢨⣿⣿⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣵⣿⣭⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢦⣿⢁⣿⠤⢤⡶⣎⠶⠾⠧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣿⣿⡿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢈⣿⣭⡖⢲⢮⣥⡍⣍⣳⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⡿⣯⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣙⣛⡳⢦⡿⢳⣠⣽⡋⠁⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⠿⠉⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣾⣿⠿⣭⡩⣙⣺⢏⣿⣡⣿⠀⣴⡇⠈⣵⣾⣿⣿⣿⡿⠿⠛⠈⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣧⠛⢋⣿⢼⣟⣿⡆⣷⣹⣿⣷⣿⣿⣿⣿⣿⣿⣶⣶⣶⣯⣿⣤⠤⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⣿⠖⣌⢻⣷⣿⠫⣿⣿⣿⣿⣿⣿⣿⢿⣟⣛⣿⠟⣿⡋⠙⠋⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣻⣮⣽⠟⣿⡄⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠉⠙⠛⠈⠁⠈⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡟⣾⢻⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⠤⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡟⠋



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢹⡦⣀⡀⢀⢀⣤⣠⣀⢀⣠⣶⢫⣿⡻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣿⡇⠋⠙⣳⠈⠁⣉⠾⠋⠀⠉⠛⣸⣿⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠯⠤⠜⢛⠀⠀⣤⢞⣨⢡⠄⡈⠙⣿⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⣦⣰⡾⠇⠀⠀⠈⠋⣄⢀⣤⢐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⠛⠟⠀⡀⠠⠀⠀⠀⠘⠿⢿⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠐⡟⠂⡤⣈⣳⣴⣤⠂⠄⠀⠀⢶⣇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⣾⠃⠸⠷⣏⣄⠀⠀⢀⣀⡤⡄⠈⡗⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⣿⠀⠠⠦⠈⠙⠶⡷⠛⠉⠢⠅⠀⢹⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣇⠀⢂⣤⢶⠖⢚⡧⠠⠰⢀⡇⠀⣸⣇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣻⣠⠀⠀⠶⠞⠉⠉⢳⣶⠇⠃⢴⡟⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣛⣯⡏⢟⣷⣶⣤⣴⣧⣀⢄⣴⠀⠹⠀⠦⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠩⣿⣾⣿⣿⣿⣿⣿⢹⣻⣄⣅⡘⠶⢦⡀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠄⠛⣿⣿⣿⣿⣿⢯⣄⣧⣌⣻⣟⣿⣔⠒⠀⠀⢙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⠒⠊⢿⣿⣿⣿⣿⢧⡹⣦⣿⡻⡃⠭⠉⠀⠀⠄⣨⡞⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣠⣦⠀⢈⣩⡿⣿⣷⣿⡎⣷⣍⢙⣉⡀⢁⡀⠀⡀⠀⡅⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡀⢉⠀⣈⠙⠿⡟⢻⣧⡇⠟⠑⠂⠉⠀⠀⠒⡒⡯⠐⢷⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠋⠐⡽⠥⢂⣄⠀⠁⢀⢀⢀⠀⠰⠄⠀⠴⠀⠁⣤⡘⢶⠑⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡦⣿⠤⠀⠀⣸⣇⢠⠄⡙⠀⠘⡆⢀⡀⠈⠊⢀⠠⠙⠦⣬⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡯⣆⠂⣠⠎⣴⣿⣾⣿⣠⣦⠎⢧⣴⡙⢀⢄⠀⠀⠀⠀⠈⠍⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⡿⢽⣧⡴⣿⡟⣟⣿⣿⢹⣿⣄⣌⣈⠓⠠⡴⢷⣴⣒⣤⡾⠿⠾⠃
⠀⣀⡀⢀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⢾⣿⣦⡲⠀⠀⠀⠀⠁⢠⣶⣾⣷⢄⡀⠀⠀⠀⣀⡀⠀⢀
⠀⠀⠀⣽⢡⠀⠀⠀⣠⣄⠰⣯⣽⠟⠀⠉⠓⠶⢆⠀⠀⠀⠈⠀⢾⡦⠻⢿⠾⠶⠶⠀⠀⠀⠀⠒⠐⠂⠀⢤⡴⣬⡤⣄⠐⠶⣧⣷⣦⡒
⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⡅⣸⣏⣤⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⣻⣾⠆
⠀⠀⣀⠠⠄⡀⠀⠀⢀⣴⣧⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢠⠈⢿⡷
⠀⠀⢸⣿⣿⣅⣠⣶⣿⣿⣿⣽⢻⣿⣿⣿⣷⣿⠃⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣴⡦⡸⢾⣦⡤⣾⠀⠘⣇
⠀⠀⠀⠈⠉⠀⠀⠙⠻⠿⣿⡂⠈⢿⣿⣿⣿⣇⢀⣀⣎⠀⠘⢾⣞⣷⣆⠀⢰⣡⣌⣆⡀⢰⣧⢀⡄⠀⢀⣰⣿⣿⣯⢊⠆⠀⢻⠛⠛⠂⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡞⠤⢿⣿⣿⣧⣾⣿⣿⣤⣠⣄⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣷⣧⣾⣿⣿⣿⣿⣷⣽⣶⡄⢀⠀⠀⠆⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⢹⣧⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠻⣿⣿⣿⣿⣧⣾⣧⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠀⠈⠸⣿⣿⢿⣿⣷⣰⡟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣯⣥⣤⣦⡤⢝⡻⣿⣿⣿⣿⡟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣷⣴⡟⣿⣦⣿⣿⢻⣧⠴⠯⣿⣿⣿⣿⣿⣿⣿⣾⡿⢱⠀⠤⠖⠈⠀⢿⣶⣧⣼⡏⠻⣿⠉⢉⣿⣞⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣲⣿⣿⣿⣿⡟⢆⠅⠀⠀⠀⠀⠈⠉⠉⠙⢿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠘⣶⣶⣿⡏⣼⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣿⣗⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠄⠀⠈⢻⣿⣃⢀⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢡⠀⠀⠀⠀⠀⢿⣿⠓⣇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢹⣿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠋⠀⠀⠀⠀⠀⠘⡟⠅⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣹⠿⣧⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⢣⠁⠀⠀⠀⠀⠀⠀⠀⠂⣄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⠿⢟⣷⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⣴⣲⡴⠂

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠿⣿⠁⣶⡎⢻⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣄⡀⣾⠉⢿⡄⠲⢤⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⣺⣕⡦⠴⠛⠻⣿⣶⣦⡝⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠮⢽⠯⢟⣫⣤⣔⠊⠙⠛⢦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣽⡞⠀⢾⣷⠄⣸⠿⣟⡠⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⢩⣿⡥⠌⣿⣇⣀⡤⣿⡻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣖⣠⣤⣖⣮⣟⣳⣽⣳⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⢈⡹⠿⡛⠯⢟⠮⠿⡷⠽
⣿⣿⣿⣿⣿⣿⣿⡿⢁⣤⣾⣿⠞⢠⣤⠤⠘⠳⣾⣧⣄⡹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠋⠉⣭⡿⢛⣻⣿⣍⠉⠙⠻⣧
⣿⣿⣿⣿⣿⣿⣿⠃⠛⢿⣿⣿⣄⠐⠿⠟⠀⣠⣿⣿⠟⠃⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣥⡀⠀⠈⢿⣯⣕⣤⣿⠟⠀⠀⣠⣼⡆
⣿⣿⣿⣿⣿⣿⣿⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢚⣿⣟⣛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⣋⣭⠤⢶⡺⠉⢠⣤⠈⠳⣌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣛⡩⠔⠚⠉⠀⠀⠀⣈⣇⠀⠈⣽⡄⠀⢿⠜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⠕⠊⠁⣀⠀⠄⠀⠀⢀⠀⠈⠘⣧⣦⣄⡐⠼⠋⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⡥⠔⠉⠀⣀⠀⠀⠉⠀⠢⡄⠁⠰⠀⠄⠀⠀⠉⠁⠀⠀⠀⢃⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢉⠰⠂⠁⠀⠀⠀⠀⢠⠀⠰⣠⠀⠎⠸⡀⠊⡀⠈⠀⠀⠀⠀⠀⠀⠀⣘⣧⠘⣧
⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⠶⠖⠛⡩⠄⠒⠈⠀⠀⠠⠐⠀⢀⠀⠀⠀⡇⠂⠀⢣⠐⡀⢣⠈⠘⡄⢀⠀⠀⠀⠀⠀⣠⠽⢿⡄⠹⣆
⠀⠀⢸⡇⢡⡥⠼⠒⠶⠦⣄⡈⠁⠀⢀⠀⠀⠀⢀⠀⢀⡁⠘⡘⡆⠄⠹⠘⡀⠈⣇⢡⠘⠶⠃⢉⡡⠀⠀⠀⠀⠀⡇⢸⣦⠹⡆⢹⡄
⠀⠀⠀⣷⠀⡇⠀⠲⣶⡆⠘⡇⠠⠀⠠⠀⠀⡄⠰⡀⣋⡀⠀⡇⢡⡌⠄⢃⣉⠕⠘⠋⢠⣀⣠⣌⠀⣀⣀⠤⠔⢒⣙⣲⣴⣞⣁⣈⣧⡀
⠀⠀⠀⠘⣇⠙⢄⠀⠼⠧⣠⣷⠂⠀⠘⡄⠠⠃⠐⣷⠀⠄⢀⡇⠀⠘⠶⠀⠀⠤⣀⠤⠛⢉⡠⢈⣈⣥⣴⠶⠿⣻⣛⣛⠛⠛⣯⠉⠁
⠀⠀⠀⢀⣘⣦⠀⠙⠒⠚⠟⠉⠀⠀⠀⢳⠀⠶⡀⢹⣇⢉⡠⠷⠊⠀⢀⡠⠔⠉⣀⠤⣊⣥⠶⠿⠿⠁⢲⣲⠟⢉⣭⡉⠳⡄⢹
⠀⠀⣠⢺⡇⢹⡎⠠⢱⠄⠀⠀⠀⠀⠀⠀⣇⡀⠱⠖⠁⠀⠀⢀⡠⠒⠁⡀⢔⣪⣷⡟⠫⠀⠀⠀⠀⠀⣈⡁⠀⢘⣻⡀⠀⣿⢸⡀
⢀⡔⠻⢸⡇⢨⢿⡀⠀⢾⡄⣀⣀⡀⠈⠀⠈⠠⠀⠀⢰⠦⠂⠁⠠⢐⣯⡶⠛⡡⠀⠐⠂⡤⠀⠀⠖⠄⠐⢷⣄⡀⡉⣄⡴⠋⠘⡇
⣾⢐⠀⠈⣧⠸⣌⣧⠀⡼⠉⠠⣤⠙⣦⠀⠀⠀⠀⣀⠓⢀⣰⣴⢾⠋⠀⢃⠰⠐⢲⠐⠊⢱⠀⣰⠃⠁⠀⠈⠙⠉⠉⠁⠀⢇⠀⣧
⢻⠘⡇⠀⢹⡀⠈⠛⣶⠀⠳⣄⠙⢂⡾⢠⠀⣂⣭⣶⠞⠋⠁⣼⠘⠂⡄⢸⠀⠀⢸⡆⡀⢸⠀⡴⡄⢰⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿
⢸⠀⡇⠀⠘⣇⠀⢘⡜⠶⠶⠶⠿⠿⠖⡟⠛⣏⠈⣷⠘⣁⣀⡇⠀⠃⠐⢸⠀⠂⡀⡇⠃⠸⠆⢃⣃⠈⠀⢀⣀⠀⠀⠀⢒⣿⡇⢺⡇
⢸⠀⢹⠀⠀⣿⠀⠀⣶⠀⠀⠀⠀⠆⠀⡧⡤⢸⢠⠟⠠⠤⠼⠧⠆⠈⠓⠈⠉⠉⠁⠙⢉⡴⠋⠁⠀⠀⠀⠀⠀⠉⠒⢎⣭⠙⢧⠘⡇
⢸⠀⢸⠀⡀⣿⡀⢀⡝⢉⣉⠑⣄⠀⠀⠀⠀⠀⠀⣄⣠⣀⣄⠀⡀⠠⠤⠤⠤⠤⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡬⠭⠥⣝⣰⠷
⢸⡀⠀⡀⠁⢸⣇⠈⣇⠀⢿⠀⣿⠀⠀⠀⢀⣀⠫⠉⠁⠀⠀⠀⠤⠤⠤⠤⢤⡄⢰⠁⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠸⡇⢣⠃⠸⠀⣷⣀⣈⣱⣰⣾⣥⣯⣤⠤⠴⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠉⠃⡆⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⣧⠘⠈⠀⡶⠭⠭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠸⡄⣣⠿⢥⣄⡤⠤⠤⠶⠖⠒⠂⠀⠉⠉⠉⠉⠀⢀⣒⣒⣒⣀⣀⣀⣈⣥⠥⠽⣦⠀⠀⠀⢸⠁⠀⠀⠇⡄⠀⠀⠴⠀⠀⠘⠂⠘⠀⠄⠀⠸
⠀⠀⠱⠏⠠⠤⠤⠴⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⢸⠀⠄⠘⠀⡇⠀⣐⠀⠀⡀⠀⢰⠀⠀⡂⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡆⠀⠁⠈⠈⠁⠙⠀⠉⠈⠉⠈⠀⠁⠙⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣶⣶⣯⣥⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⣀⠀⠀⣀⣀⣀⣀⡤⣤⣠⢤⣄⣀⣀⣠⣤⣤⣤⠀⡔⡖⣿⣱⣽⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡼⠿⠘⠁⠀⠓⠈⠐⢛⠛⣫⠏⡇⢟⡕⠙⠟⠛⠉⠉⠉⠉⠉⠁⠉⠩⠤⠀⠉⣼⣯⣿⠅
⠀⠀⠀⠀⠀⠀⢀⢔⡕⠁⠀⢀⢤⡠⡀⠀⠀⡰⣡⣾⡟⡼⢱⢿⡇⠀⡆⠀⠀⢠⣠⣤⣤⡄⠀⠀⠀⠀⡇⢩⣏⣿⡄
⠀⠀⠀⠀⠀⡠⡣⠋⠀⠂⠀⠀⠂⠑⠁⠀⠀⣿⣿⣧⢊⡆⣿⣿⡇⢠⠃⠀⠀⠊⠛⠛⠛⠀⠀⢀⠀⡜⡀⣾⣿⣷⠃
⠀⠀⠀⠀⡔⣵⣅⣤⣤⢠⢂⡀⡴⠴⢂⣀⣀⣰⡿⢃⢪⡇⣿⣫⣄⣁⣀⣀⠃⣀⡀⢃⣐⣛⣫⠭⡜⣸⡞⣼⣿⠃
⠀⠀⠀⢬⣬⣭⣿⠿⣶⠶⣶⣮⣭⣥⣭⣍⣙⣻⣣⣇⣬⣇⣿⣭⣿⣿⣿⣿⣿⣿⣿⣯⡭⠼⠶⠶⡾⣻⣿⣿⠁
⠀⠀⠀⢸⢻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣗⣒⣶⠿⣿⣄⣀⣈⣭⣙⣶⣖⣶⣿⣿⠇
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⢯⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣿⣦⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⢀⣤⣄⣀⣀⡴⢦⣀⣠⣿⡿⠛⠉⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⣠⠞⠁⠈⠉⢁⣠⣶⡿⠟⠉⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⡾⠁⠀⠀⣠⣴⣿⠟⠉⢀⣠⠶⣤⣀⣀⣠⡄⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⡴⠋⠀⣀⣴⣿⠟⠋⢀⣠⠶⠋⠁⠀⠀⠉⠉⠁⠹⣦⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⣀⣴⣾⠿⠋⠀⢠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣤⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⣤⣾⡿⠛⠁⣀⠀⠀⢻⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣠⣶⡿⠛⠁⢀⣴⠞⠉⠀⠀⣼⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡿⠟⠉⢀⣤⠞⠋⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⢠⣾⣿⣿⠟⠉⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢀⣠⠶⠋⠁⣠⣴⡿⠟⠉
⠀⠀⠀⣰⣿⠋⢸⣿⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣤⣾⡟⠀⠀⠀⠛⠁⣠⣴⣿⠟⠋
⠀⠀⠀⢿⣧⡀⠀⠹⣿⣄⠀⠠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠈⠻⢿⣦⣄⠘⢿⣧⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠀⢀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣌⠻⣷⣄⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠙⠛⠉⠙⠳⣦⠞⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣆⣀⣠⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⢛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠿⣷⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣄⣀⣤⣤⣴⠶⠛⠋⣀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣁⣠⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣾⠿⠟⠛⠻⠿⣶⣤⡐⢶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡉⠙⠛⠻⠿⠿⠿⠏⣰⡿⢋⣤⣶⡿⠿⢿⣶⣄⠙⢿⡆⠹⠿⠿⠟⠛⠋⢁⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣤⡄⢸⡿⢡⣾⣿⡏⢰⣶⣦⣽⣿⣷⡀⣴⡄⢠⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⡇⣿⡇⢸⣿⣿⣷⣬⣈⡙⢿⣿⣿⡇⠘⠃⢸⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣦⣤⣤⣤⣄⢸⣷⡈⢿⣿⣏⠻⠿⠟⣸⣿⡿⢁⣿⠇⣀⣀⣤⣤⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠻⣷⣌⠛⠿⣷⣤⣾⠿⠋⣠⣾⠏⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣧⡌⠛⠿⣶⣦⣤⣴⣶⠿⠛⠡⠾⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⡄⡀⠀⠈⠀⢠⣤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢠⣶⣿⡇
⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⢠⣿⠆⠀⠀⡀⠈⢀⣄⠐⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⡀⠆⠀⠀⠈⠙⡀⠠⠐⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⡔⠀⠠⢠⠂⠀⠀⠀⠐⠪⠋⠀⠀⢀⣀⠀⢀⣀⠀⠀⢠⡄⠀⣰⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⢻⠊⠀⠀⠀⠄⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠸⡜⠓⠀⠀⠀⠀⠈⣹⠁⠐⠀⠀⠄
⣀⠀⠀⠀⢠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠸⠀⠀⡄⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀
⡋⠀⠠⠄⠉⠉⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠈
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠐⢀⡀⣠
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢀⣀⠀⠀⠐⠀⣀⡀⠀⠈⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠈⠙⠀⠀⠈⢸⣷⡀⠀⠀⠈⠁⢀⡀⠄⣧⢀
⣀⣤⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠀⠀⠠⠀⠀⠀⠘⠁⠀⣿⣿
⢈⠙⠂⠀⠰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠋
⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠴⠀⠀⠀⠀⣿⠠
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⢱⡇⣀
⠆⠀⠀⠀⣀⡳⠀⢠⠄⠀⠀⠀⠀⠈⢠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣾⠁⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠐⠄⠀⠀⠁⠀⠙⠋⠀⠀⠀⠹⠟⠃⠀⢀⠀⢀⣦⠠⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠐⠋
⡀⠀⠰⠆⠀⢠⡇⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⣠⡤⠃⠀⠀⠰⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⣤⡄⠉
⠁⢀⠀⠀⠀⢹⡇⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠈
⠀⠀⠄⠀⠀⠘⠇⠉⠀⠄⠀⠀⠐⠐⠀⠓⠀⢀⡀⢰⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⡆⠀⠀⠀⠀⠀⠈⠂⠀⠄⠐⠀⠀⠀⢀⡌⠀⣼⣿⣾⣇⢀⡀⠀⠠⢀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠈⠁⠉⠉⠙⠉⠀⠀⠂⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⠛⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⠂⡀

⠀⠀⢀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣄
⠀⠀⣿⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⢀⡴⠾⠿⢦⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⣄⠘⢠⡇⢠⣤
⠀⢐⣿⣿⡏⠐⠘⣿⣟⡃⠀⠀⢀⣿⣿⠇⡸⣿⣿⡄⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠉⠀⠒
⠀⠀⠸⣻⡇⠀⠀⣿⠷⠀⠀⠀⢸⣧⣀⣤⣤⣤⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀
⠀⠀⠀⠘⠳⠶⠶⠷⠀⠀⠀⠀⠸⣿⣿⠿⣿⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢛⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢺⣿⣷⣶⡶
⠀⠀⠀⠐⠒⠒⠒⠚⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠛⠁
⠀⠀⠀⠀⠀⢀⢀⠀⠀⢀⠨⠵⠀⠀⠀⡆⣴⠀⣶⢰⣴⢰⡆⢰⢰⣴⢀⣦⢰⡀⡦⢠⡆⣴⣴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠬⡿⢗⠂⠀⠈⠉⠀⠀⠀⠀⠁⠉⠀⠘⠈⠉⠈⠁⠈⠈⠈⠀⡉⠈⠀⠁⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠁⠁⠁⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⢩⣦⣤⣴⡟⠁
⠀⠀⠀⠛⠐⠛⠂⠺⠁⠀⢀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢴⣤⡿⠼⢿⣥⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣭⢻⣿
⠀⠀⠀⢠⣀⠀⢀⣀⠀⠀⠈⣿⡆⠀⠀⠀⣿⠇⠀⠀⣴⣦⡇⠀⠠⠀⠹⣥⣦⠀⠀⠀⠀⢸⣷⡦⡀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⢐⣀⣈⣩⣭⠀⠀⠀⠘⠇⠀⠀⠰⠟⠀⠀⠀⢶⣾⡇⠀⠀⠀⢀⣷⡶⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠸⡇⠀⢸⡟
⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠱⣿⢻⣶⣟⣿⡆⠁⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⢸⣶⣾
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⡋⠀⠀⠀⠀⠰⡶⠶⠀⢨⣽⣥⡀⠸⠿⠷⠀⠀⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






⡿⢿⡿⠿⣿⡿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣟⣛⣿⣿

⠀⠀⠀⠀⠠⠦⡀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⡿⡿⢦⠤⣾⢱⣏
⠀⠀⠀⠀⠀⡦⠁⣤⠀⠉⠋⢹
⠀⠀⠀⠀⠈⠇⣄⢋⡿⠷⡀⢈⣆
⠀⠀⠀⠀⠸⣇⣿⣧⠀⠀⠄⢸⣧⣆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠈⠀⣽⣿⠶⠤⢶⠖⠒⠒⠀⠐⠒⠒⠦⠤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣴⣿⣿⣦⣄⠀⠐⡀⠀⠀⠀⠂⠂⢄⠁⠾⣿⣿⣿⠿⠷⡶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢚⣿⣼⣧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡷⢾⣿⣿⣗⣙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⣬⣹⣉⣧⡀⠀⠀⠀⣀⣀⢔⠀⠀⠀⠀⠘⠟⢿⣽⣿⠏⠾⢣⢴⣬⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣤⣵⣾⣌⣿⣿⣿⣿⠿⠿⣯⣿⠀⠀⠀⠀⠀⠀⠂⠀⠈⠙⢿⣝⠀⡰⣍⠂⠉⣭⠿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠈⣿⣿⡟⠀⠀⠀⠀⠀⠙⣧⣷⣦⢁⣀⡈⠀⠀⠀⠀⠀⠈⠳⣧⣀⠾⣿⢋⠒⣹⣿⣿⣻⡖⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠻⣿⣿⠻⣿⡳⡄⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣈⠛⣿⡀⠉⣿⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⢸⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠶⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣀⣼⡏⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠾⠿⠃⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠚⠛⠃






⣿⢿⠿⠿⣿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠃⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣾⣿⣁⡀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣾⣶⡿⠿⠛⠛⠛⠛⠛⠛⠉⠙⠉⠉⠉⡆⠀⠀⠀⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠙⠻⣟⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⠀⢀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⢩⣍⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣰⣿⣏⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠻⣿⣯⣿⣿⡄⠤⠀⠀⣀⡤⠤⣄⠀⠠⠤⠤⠄⣾⣿⣿⣿⠛⠿⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠈⠛⣿⣿⡅⠀⣶⣦⠀⠀⠀⣠⣴⣾⣷⣦⡀⣿⡟⠙⢯⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠛⠒⣿⣿⠧⣼⣿⣧⣤⣤⣾⣿⣯⣍⣩⣿⣿⣿⣷⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠇⠈⢿⣿⣿⣿⠿⠁⠉⣉⣽⣿⠏⢽⣿⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣀⣀⣀⣉⣉⣡⣤⣤⣀⠈⣛⣁⣀⣿⣿⡄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⠉⠁⠀⠀⠀⢸⣇⠀⠉⠛⢤⣀⠀⣹⢹⣧⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⡀⠀⣰⠞⠶⣀⢿⣄⠀⠀⠀⠈⠉⠻⢀⣽⣇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠴⠶⢾⣿⠇⠀⣿⣦⠀⠈⠃⠙⣷⣦⣀⠀⠀⠀⠀⠙⣿⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣿⣄⠀⠀⠈⠆⠘⠏⠳⢤⣀⣀⡄⠘⣷⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢷⣤⣤⣤⣿⣶⠀⠀⠀⠐⢶⣿⣾⣿⣻⣿⣦⠈⠇⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠁⠙⢏⡀⠠⣧⡀⠀⠀⠀⠙⢿⣿⣿⡙⢧⣄⠈⠀⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠁⠺⣿⣿⣷⣄⠀⠀⠀⠈⠛⠷⣄⣽⣧⠀⢠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⢀⡀⢀⠈⠉⠛⢿⣷⡄⠐⠀⠀⠀⠀⠋⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢠⣾⣿⣧⣄⠀⠀⠀⠉⠑⠲⣆⡀⠀⠀⠀⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣾⠿⣷⠻⣿⢿⡄⠀⠀⠀⠀⢻⣅⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠁⠀⢀⢤⡀⠀⠀⢠⢜⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢲⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣤⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠿⢿⣿⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠠⠴⢬⣭⠷⢶⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠳⢦⣶⣦⣀⠂⠠⠍⢸⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠙⠓⠚⠂⠿⠿⠿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣄⣀⣀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣛⡿⢿⣿⣿⣿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛⢛⠹⠁⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⢿⡕⣃⢹⢰⣿⠷⠲⣦⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⣿⣅⠀⢿⣟⠀⢶⡏⣸⢛⡿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠌⣻⣶⠋⠉⢛⣾⡷⣷⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⢳⣦⡘⠦⢸⣇⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢋⣤⣠⣾⣟⠛⢿⣿⠉⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣴⣶⡤⣤⣞⡃⠀⠀⠀⠙⣿⡧⠾⡛⠛⠛⠚⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣻⣿⣿⣿⣿⣿⡇⢿⣿⡃⠀⠀⢀⣼⠟⠳⣶⣿⣶⣤⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⡇⢿⡋⢡⣄⡀⣀⠏⠀⣧⣉⣰⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠈⠉⣿⡃⢀⣾⢿⣿⡛⢻⠉⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣿⣿⣿⣿⣿⣿⣿⣇⢸⣇⣀⣴⣿⠂⠀⣼⡿⠈⣷⠈⢲⣶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⢿⣿⣵⠃⠀⢰⣿⠃⠀⠉⠄⢡⢻⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⢃⠎⠀⡼⠂⠀⠀⠀⠀⠈⢆⢻⡍⢌⢷⣀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⡿⢡⠏⠀⣼⣤⡶⢾⡷⠶⣤⣄⠈⠊⢿⡌⢣⡻⣷⣄⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠛⢰⠋⠀⣴⣿⣿⡀⢸⡇⠀⠈⢻⠀⠀⠈⢻⡄⠀⢹⣾⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠈⠃⠀⠀⢻⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣹⡇⢀⣤⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡌⢿⣕⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣿⠀⣞⠇⡐⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⣷⢀⠀⡄⠀⠀⠈⣿⣿⢞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⢸⣿⣿⡃⠀⢠⠀⣄⠀⠀⠸⣿⠀⢻⣿⡟⠈⢀⣡⢀⠁⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣾⣿⣿⣧⠀⣞⡀⢿⣷⣤⣠⣿⣤⡿⠋⠀⠀⢿⡟⢸⣼⣈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢠⣿⡇⠀⠀⠀⠉⠛⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠀⣿⣿⢳⣼⣇⢠⢰⢠⣰⢠⣧⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣧⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠑⠂⠤⠀⠀⠀⢀⡀⡀⠀⠀⠀⣀⠀⢀⡀⠠⠤⠀⠀⣀⣀⠀⠀⠄⠊⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣂⣀⠑⢦⣤⣀⢄⡀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⠤⢆⡟⠁⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠙⠳⣤⣤⣄⣉⣉⡚⠓⢦⡠⠀⠀⠔⡛⢛⣈⣉⣴⠾⠋⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠲⢤⠔⡈⠄⠛⠉⠙⠋⢍⠻⣇⡴⢞⠟⠛⠙⠉⠈⢝⠳⢮⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡏⠠⠁⠌⠀⠀⠀⠀⠀⠀⠀⠑⡈⠡⠂⠀⠀⠀⠀⠀⠀⠐⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⢺⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣽⠁⠆⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣧⡄⠐⡀⢆⠀⠀⢠⠟⠁⣈⠑⣾⣄⣳⣆⠔⡉⠓⢄⠀⠀⢠⠂⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢳⡄⠐⢌⠳⣄⣹⣀⠀⣉⣼⡇⠀⢸⣷⣄⡋⡀⣼⣀⠔⢁⠀⡼⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣾⡄⠀⠈⠳⣄⡁⠒⠊⠭⠻⠽⠍⠘⣤⡟⠿⠿⠹⡋⠡⠄⠐⣁⡖⣠⡟⡀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠒⢶⢶⣛⠻⢿⣶⣾⣶⣤⡀⠀⠈⠑⢤⡤⢤⣤⣀⣀⢄⠁⠝⠶⣤⠤⠦⡤⠴⢊⠻⣈⡤⣿⠮⠓⠈⠃⠉⣀⡤⠛⢒
⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠈⣉⠓⠶⠭⢧⣉⠐⠛⠙⠒⠓⠲⠦⠦⢤⣤⡀⠀⣠⣰⠶⣦⣶⡶⠶⣯⡷⢟⣩⣄⡦⠥⠶⠛⠉⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⡠⠔⠂⢉⣠⠴⠾⠀⠀⠀⠀⠀⠉⠑⠒⠰⢖⡤⢤⣉⢶⠟⠙⡆⠰⢈⠛⣁⡠⢤⠳⠮⠛⠋⠉⠁⠀⠀⠰⣤⣀⣀⠈⠁⠓⠠⣀
⠀⠀⠀⠀⠀⡎⠀⠂⢈⣁⠴⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠂⠐⠳⠶⠶⠶⠶⠦⠵⠈⢉⠴⠒⠂⠀⠀⠀⠀⠀⠀⠈⢥⣀⠁⠉⠑⠢⠀⠈⠑
⠀⠀⠀⠀⡔⠁⠀⢋⣉⠴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡉⡃⠓⠦⠀⠀⠇
⠀⠀⠀⠀⠈⡁⠘⢋⣤⠞⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⡁⠒⠦⢀⠊
⠀⠀⠀⠀⠀⠈⣄⣻⠅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⡀⠀⠠⠀⠀⠀⠈⠀⠀⠂⠀⠙⢲⣄⡠
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⡎⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠈⠀⠐⣀⣠⣤⠶⢂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⠈⢻⣐⡒⠤⠄⢀⢀⡀⠂⢠⠀⠀⠀⠀⠁⠘⠐⢀⡠⠠⠒⠈⠉⢃⣴⠎⡠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠒⢌⠑⠤⡀⠀⠀⠉⠈⠉⠣⣤⡀⣬⡤⠖⠈⠀⠀⠀⠀⣤⠴⠫⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣀⠤⣚⣁⣎⠂⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⢖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡀⠰⠗⣼⠻⣿⢋⡴⠞⢋⠍⠁⠁⠋⠃⠁⠐⠐⠒⠿⣿⡷⣿⢿⣿⠂⠀⡁⠂⠂⠀⢀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠉⠀⠛⠁⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠙⠛⠃⠀⠂⠀⠀⠀⠀⠀⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠛⠙⣿⠁⠁⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣀⠀⡄⠀⢻⠁⠀⢂⣀⣼⡦⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠙⠻⠷⣔⣾⣷⣤⣷⠾⠛⠉⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢺⣷⡜⠻⠿⠟⢣⣴⡟⠂⠀⢸⠀⠀⠀⠀⠀⠀⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠁⠀⠐⣤⡁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣀⣤⣾⣿⣷⣄⣀⠀⣀⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣮⡙⠉⠸⠿⠁⠿⠋⠁⡉⣿⢿⣟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣄⢀⡁⠓⠀⠆⠠⠂⠀⠉⠀⡠⣾⣿⣣⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠠⠀⠉⠀⠑⠀⠄⠤⠰⠀⠊⠀⠀⢿⣿⡻⣏⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠳⠠⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠁⠀⠸⣿⣿⡯⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠶⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡴⢿⣿⡢⢎⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⡌⠀⠀⡀⡀⡀⣀⠀⡄⠀⠀⠀⢀⢀⣾⣿⣿⣟⡳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣧⣌⢘⢇⠟⠸⢏⡾⣳⡟⠻⣃⢀⡄⢰⡿⣿⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢸⣿⢛⡷⣿⢼⣧⠾⠂⢿⡀⢡⢻⡴⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⣿⣘⣿⠾⣣⢿⣿⣹⣿⠘⣛⡘⣩⣤⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⣟⢻⣿⣿⣿⣿⡇⣹⣯⢋⣷⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⢿⣿⣿⣿⡼⣿⣼⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢾⣿⣿⣿⣿⣿⠻⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣷⠏⠉⠻⣿⣿⣟⣻⣿⣯⣴⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣾⠃⠀⠀⠀⠈⢻⣯⣽⣿⣭⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣾⣅⣼⠀⠀⠀⠀⣠⣦⣦⣼⣏⣿⣟⣾⣟⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⣀⡹⡟⢻⠹⣧⣀⣀⠀⡣⣿⠙⣟⢩⣤⣀⣄⣈⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠉⠉⠑⠦⢿⠗⠏⠉⠁⠘⠟⢿⡶⠿⠊⠈⠙⠿⠿⠿⠉⠉⠉⠉⠁





⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⡈⣀⣀⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⡤⢤⡀⠀⠀⡀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡄
⠈⠳⠞⠁⠀⠀⠀⡉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⣿⡤⡆
⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠇⠀⣀⠀⢸⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄
⠀⠀⠀⠀⠙⠃⠀⡿⡿⣿⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠿⠟⡤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⡼⢿⣿⠀⣰⠂⢰⢶⡄⠀⢀⡄⠀⠀⠀⠀⠀⠀⣮⣆⣀⠤⡀⠐⠒⠂⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠃⠀⠈⠙⠉⠁⠀⠀⠀⠁⠈⠀⠀⠀⠈⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠄⢰⢶⢤⡀⢀⣤⣤⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠈⠺⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠀⠒⠐⠐⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⣴⡶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⠿⣮⣵⠱⠿⢿⡚⠻⣅⢉⢳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠽⢟⠻⣉⢒⣖⣀⣌⣃⡸⡿⢾⠟⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⡶⢆⣙⣦⠉⠉⠙⠉⠉⠁⣸⣠⣄⣴⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⢭⣏⢻⡍⢵⠆⣀⡀⠀⠀⢠⣿⣿⣟⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡗⣦⡈⣐⡺⣙⠫⠝⢧⡔⡈⠉⠉⠉⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣟⡵⣏⡳⢿⡻⠬⢷⠴⣖⣷⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡳⣭⣛⣶⠥⣮⡉⣖⣒⣶⡢⢦⣾⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣱⢽⣵⣪⣟⣛⣿⢟⡽⣯⢴⡯⣸⣿⠈⠏⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠨⢙⡒⠯⣶⢶⣚⣫⢗⠛⣺⠧⢋⣿⡆⣺⢊⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠈⣯⠙⢒⡙⠉⡠⢗⣐⠏⡴⢟⣞⠛⣿⠦⠖⡎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡢⠝⠲⣂⣸⣏⣭⠔⣋⡴⣚⠝⣽⢿⣟⢻⡕⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢟⣉⣠⢒⣸⠦⠶⠋⣁⡸⢵⡎⡼⡾⣿⣿⡻⣷⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡬⠴⠣⠤⡥⠒⠣⡭⢖⣫⡞⡴⣣⢼⣿⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣔⡗⣓⠿⢟⣩⣖⡾⠃⢞⣩⡦⠋⠻⡿⣷⡻⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣶⣬⣔⣛⠲⣦⠬⠭⠭⢖⣞⡽⢛⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠏⠈⠳⠷⢹⣆⠸⣭⣭⠵⣶⣪⡴⣻⣿⣿⣿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣤⣤⣱⣏⠐⠚⡉⠛⠻⠬⡠⢼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡂⣠⣶⣫⡞⠁⠀⠀⣽⡶⠢⣯⠂⠤⢤⡀⢲⡹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣰⢭⣙⢀⣤⡴⢻⠁⠀⠘⢷⣤⣤⡈⢄⡼⣝⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣙⠂⠈⣿⣿⣯⣛⠢⢄⡀⠘⣿⠀⠙⠀⠁⢸⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠳⡌⠙⠲⠇⠀⢿⡄⢀⡴⠀⠗⢧⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⢸⣷⠀⢿⡂⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⠀⠀⠀⠀⠀⠀⠈⣿⡄⠈⠥⡀⡲⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⡆⠀⠀⠀⠀⠀⠀⠛⠉⢒⢀⣇⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠳⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡽⣆⢀⡑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⡀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣹⡆⠉⠀⠈⠛⠛⢲⡆⠐⣆⢰⡀⣦⡀⠉⠀⠀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣆⠀⠀⠀⠘⠂⢉⢶⢉⡏⣥⣡⠀⠀⠀⢠⡿⢿⡇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⢃⣀⠀⠀⠀⠛⢂⠬⢵⣤⠟⢀⣀⣴⠁⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣸⡿⠷⢦⣄⠀⠈⣥⠐⢁⣴⣾⣋⣩⣦⠀⠀⠈⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠇⠀⠀⣿⠉⣿⡷⢻⣷⢾⣿⣷⣾⣯⣙⡿⠃⣼⠃⠀⠀⣿⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⣽⣤⣬⠀⠞⠻⣾⠁⣹⡈⢸⡟⠗⠶⠒⠀⠀⢰⣿⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠹⣧⣄⣄⣄⢀⠀⠀⣄⣴⣿⡆⣿⣷⣦⣤⣀⣄⠀⠀⠀⣾⠇⠀⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢰⡷⠈⠹⣋⢩⠷⣿⣿⣿⠟⢿⣟⡛⣿⠛⡉⣁⠠⠀⣛⠃⠀⠄⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⣡⣔⡿⠛⠋⠾⠏⣠⡟⠁⣞⠨⠑⡬⢀⠀⠈⢠⣾⣿⠷⣰⣆⠙⠂⠉⡷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣽⡷⠆⢠⣄⠀⠀⣶⠀⢁⡤⠀⠂⣼⡧⠀⠀⢹⢏⣓⣤⠙⠘⢷⡀⠀⠼⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⢠⣿⣿⡴⣾⠞⠀⢠⣿⢀⠈⣿⠘⠓⢘⡿⡠⠀⠘⠿⡓⠸⣥⠀⣦⢀⠀⢤⠀⠿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣼⠉⡟⢡⠿⠄⠀⢼⣛⠘⡀⠀⠀⢰⣅⠀⠴⠀⠀⠀⠑⠤⠛⠀⠀⠈⣀⠀⢤⡀⠀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⡷⢮⡻⢿⡀⢠⣶⠈⣿⣣⠈⢠⡀⡈⢿⡎⠚⢵⣄⡀⠀⠀⠃⠀⣤⡀⢻⡄⡞⠗⠂⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠐⣶⣵⠜⣷⠈⠉⠀⢻⡏⠁⣴⢃⠷⠀⣀⠷⡈⣿⠧⠀⠀⠀⣃⠈⠗⠻⠄⢲⡄⢀⢢⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢻⠗⡈⠻⡗⠀⠀⠀⢡⡦⠈⡉⢠⠀⠹⣆⣨⡄⠀⠘⢧⡀⢋⢠⣄⢲⣴⣄⢀⠘⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢞⡠⠈⠁⠀⢠⡀⡀⢀⠀⠂⠰⡟⢈⣆⠀⢟⣷⢦⢀⡲⣼⣿⠀⠃⠈⢷⡆⢛⡘⠃⢀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣆⠈⡧⠀⢲⡇⣼⡇⠀⢰⣆⠊⢽⡄⠀⠡⠀⣿⠁⡀⠉⠀⠀⢂⠀⢳⠼⣧⠱⠆⠛⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢋⣿⡆⢽⣧⠙⠀⠀⠈⠋⠀⣨⡿⢤⠐⢢⣿⣧⡘⠃⢙⡀⠞⢿⣟⣦⣶⣀⡴⣜⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⠉⠾⠟⡁⠀⠀⣆⠀⢀⡋⠐⠏⠄⣤⠉⠉⠩⡄⠀⢷⡇⠈⣿⣿⣭⣤⢃⣉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄⢄⣻⣦⠀⢹⡆⠸⣧⢦⡤⠠⡜⢧⣀⠂⣿⡀⠉⠃⠀⢸⣿⠛⣙⣟⢻⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡎⣫⠹⠇⠀⠻⠀⢹⡴⠻⣆⡘⠚⠛⠀⠈⠁⠀⠀⣡⠰⣦⣷⢟⣅⣱⡕⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⡄⠰⡄⠀⠀⠲⢑⣧⣻⣦⣳⣆⠘⣦⡘⠀⠀⣿⢐⣿⢥⣾⡿⢉⣼⣯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠜⢢⣴⣬⡤⠗⠛⠛⠙⢦⣽⡏⠛⢞⠙⣧⠀⠀⣼⣿⣾⢿⣾⣾⠏⣼⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⢋⣠⡆⠼⢿⢛⢁⣤⠊⠀⣄⣄⠘⠣⡀⠈⠛⠻⣶⣀⣠⡍⢀⠿⢢⡿⠛⠿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣸⡿⣄⠐⠁⠀⢰⣿⡽⠋⣤⢀⠏⠀⠁⢽⡦⣴⠀⠀⠸⣿⣏⡿⢸⠘⠂⣾⣽⣷⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠸⡏⠉⠈⢦⠀⠀⠀⠀⠈⠀⠀⠀⡟⠋⠀⠀⠀⠀⠐⠋⠀⠀⠀⠙⢿⣿⡸⢼⠀⣿⠇⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀⣤⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⠀⠈⣿⢿⣷⣾⣿⣿⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⢁⣤⡦⠟⠛⠋⠉⠉⠀⠀⠈⠙⠿⠦⣄⢀⠀⠀⠀⢠⣴⡾⣧⠀⠸⣿⣿⣾⣿⣿⣤⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠀⢀⣶⣿⡿⠇⠀⠀⠈⠙⠷⠍⠎⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣦⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠱⣎⢹⡎⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⣿⢳⣄⠀⠸⣴⣗⣏⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⢀⣤⠀⡀⠀⡤⢺⣟⡏⠸⣆⠉⠳⣤⠀⠈⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢿⡀⢸⣾⠁⣼⡇⢸⠇⣼⠃⢿⡀⢻⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡈⢷⣼⡇⢰⣿⡇⢸⡄⢿⠀⠀⠹⣄⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠠⣝⡻⣮⠇⣧⢸⡀⢧⡘⢧⡈⠳⣛⢿⡆⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⢀⣉⢙⡚⠂⠀⠈⠁⠀⠙⠀⠉⠀⢉⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠨⠍⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌
⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⣀⠀⠀⠀⠀⠶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁
⠀⠀⠀⠀⠀⢀⣾⡗⢢⢹⡆⣽⢰⡆⣀⡀⠛⠿⠃⠀⡀⠀⣄⠀⢠⠄⠀⠀⠀⠛⠛⠂
⠀⠀⠀⠀⢀⣾⡟⣧⠾⠚⠱⠋⢠⡇⢹⡇⣶⠐⣶⣦⣷⢀⠈⣧⡿⠀⢠⡇
⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠠⣴⡿⢁⡿⠁⣿⠀⢹⡎⣿⠘⡆⢹⣇⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠞⠁⣼⠏⠀⢸⣇⣸⠀⣿⢸⢿⣴⣶⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢀⣤⣿⣦⣿⠀⣿⠾⣾⣩⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣶⠞⢛⣥⣤⠬⠟⠀⠁⠀⠈⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⠀⠈⣶⣖⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠩⣾⠀⠀⠀⠀⠀⠠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠿⡆⠀⠀⠀⢀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⢸⡧⠀⠀⠀⠈⠻⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠀⠀⠀⠾⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠿⡻⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⠉⣨⣴⣿⣿⣿⣿⣿⡟⢿⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣷⣶⣼⣾⣷⡟⢩⣿⣿⣿⣛⢛⣭⣿⠎⣉⡠⢧⡃⠉⢻⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⣟⣟⣭⣭⣽⢻⣬⣷⣾⣿⣿⣿⣿⢿⣿⣿⣿⣭⠤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠞⠁⢁⣬⣶⣿⣿⣿⣿⣿⢿⣯⢿⣿⢿⣿⣿⣦⡥⢿⣿⢿⣿⣿⣿⡿⠒⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⡟⣼⣿⣿⢿⣹⣿⣟⣻⣿⢹⢿⣿⣿⣿⣿⢸⢻⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠷⣼⣿⣿⣿⣆⣻⣿⣿⣖⢿⡜⠘⠏⢹⣿⣿⣌⡀⠀⠹⠛
⠀⠀⠀⠀⠀⣼⠿⠟⠏⠈⢠⣿⣿⡝⠯⡧⢹⣿⣯⡓⠸⣷⠀⠀⠏⢿⢿⡓⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡿⡝⠆⠀⢸⣿⣦⡉⠀⣿⢀⣤⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠑⠈⠀⠀⠀⠻⠇⣩⣶⣿⡝⢿⣿⡿⣶⣤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⡿⣫⠵⢿⣹⡇⠀⠀⠹⡄⢹⠷⠄⠀⣀⣙⠿⣔⣲⣤⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠛⠛⠛⣿⣷⣾⣖⣲⣤⣄⣿⡗⠛⠛⠓⠂⢲⣶⡼⠶⢦⣸⣷⣶⡿⠿⠿⠿⠛⠛⠛⠻
⠀⠀⠀⠀⠛⠉⠛⠒⠒⠒⠚⣁⣤⣤⠤⠤⠌⠈⠉⠒⠒⣿⡇⠉⢁⣷⡆⢠⠈⣡⠈⣋⣉⣣⠤⠴⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⡾⢸⢻⢃⣮⢞⢥⠀⠠⠤⠭⣭⣿⣿⣯⣈⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣬⡏⢀⡟⣯⣈⣬⡃⢻⣾⣯⣀⠀⠐⣾⠿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠒⠊⠉⢛⡽⠋⠋⠀⠸⠘⠹⠉⠳⠈⠙⠛⠿⠿⢽⣿⠿⠿⠓⠒⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⢈⠙⠿⢿⣶⣆⣰⣶⣷⡟⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢣⠞⢩⣿⣄⠀⠉⢛⣿⡿⠋⣠⠼⠇⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡷⢀⣶⡆⣼⣿⣶⠈⢘⠁⣿⡯⡷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⣈⠂⣉⠸⠞⠋⢀⡿⢇⠙⠚⠁⢀⡾⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡻⣿⣓⡀⠀⢀⣼⣿⣧⣾⡂⠀⠀⡿⠁⢂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⡉⢁⣁⣤⣤⣌⡉⢋⣠⣤⣴⣤⣴⣿⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⢿⣿⠿⣻⣿⣽⣿⣿⣟⣿⡍⢿⣤⡐⠲⣴⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣽⢿⣽⣿⣽⣻⣿⣿⣻⣿⣿⢻⣦⣍⣻⣷⣶⡝⢿⣗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⠙⢙⣿⣿⠟⢉⢯⣿⣿⣿⣽⣿⣿⣏⡬⢿⣽⣾⠁⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⠀⢠⣈⠘⡈⠁⣄⠙⠿⠿⣧⡹⣿⣷⣟⣿⡹⣿⠙⣿⣶⣤⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⢀⠹⡄⠀⢄⠈⡁⠙⠷⢮⣿⣮⡻⣿⣿⣷⣻⣄⢄⠉⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠑⣻⡄⡤⢤⠙⢦⡠⣄⠉⠛⢿⣎⠹⣿⣿⡸⡆⢧⠀⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⢦⡈⣿⡜⠈⠳⡄⢃⠙⢶⡐⢦⠙⣷⣻⣿⣷⣹⠘⠣⡈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡙⠷⣝⢿⣦⣾⣧⠀⢧⠈⢧⠀⡉⠺⣷⣽⣯⡛⢷⡷⣦⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠉⢻⣤⣙⣦⢻⣿⣿⣰⢸⡌⢮⡳⠘⢷⣤⣿⣿⣿⣷⣿⣤⣹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠻⣿⣻⣷⣿⣿⣿⣈⣿⢆⡛⢾⣦⣉⣙⣝⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠹⣿⣿⣿⣿⣽⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⡿⠋⠻⣿⣧⣿⣿⣿⣿⣿⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⡝⣇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡿⠟⠁⠀⢀⠀⢻⣿⣽⣿⣿⣿⣿⠄⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⠾⣿⣦⣼⡇⠀⠀⠀⠀⠀⠀⣾⢠⡿⠿⠃⣰⠟⢀⣾⡿⠛⠺⢿⡿⣿⣿⣿⣿⠀⠀⠈⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣀⣰⢞⣿⣇⢢⣀⡀⠄⣐⡶⢢⣾⣿⣾⣏⣁⣿⣿⡄⣾⣿⡀⢁⠀⢀⡈⠉⠚⠛⠫⣀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠙⢿⣷⣮⣿⣧⣼⣥⣾⣶⣿⣺⣿⣿⣿⣿⣿⣿⣶⣷⣿⣷⣭⣓⠷⣤⣅⡀⠀⠀⠀⠀⠀⢒⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣭⣭⣽⣟⡻⣝⣛⣿⣿⣿⡿⠿⠿⢿⡿⠿⡿⠿⠿⢿⣿⣿⣿⣿⣭⠄⠰⠂⠢⡄⠀⠀⠀⠀⠀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠝⠻⣿⣡⣼⣿⣯⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⡄⢠⡶⢀⡄
⠀⠀⠀⠀⠀⠀⣤⠅⠀⠀⢀⠛⠃⠀⠀⠀⢸⠧⣿⣇⡞⢀
⠀⠀⠀⠀⠀⠘⢁⡀⠘⢶⡾⢦⠀⠀⠀⠀⣼⢀⡇⢹⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⡀⠀⠈⠀⢀⣹⣿⡄⠀⠀⠀⠀⠻⠏⠓⢾⢆⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⢠⠀⠀⢠⠆⠀⠉⢟⠀⠀⠀⠀⠀⠘⠃⠀⠀⠙⢻⣿⠇⠀⠀⠀⠀⢀⡤⠊⠀⢀⡤
⠀⠀⠀⢀⠀⠀⣼⠃⡀⠀⡍⠀⠀⠀⠀⠀⠀⠠⠀⠈⠈⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣾⠃
⠀⠀⠀⠈⠀⠱⢼⣀⡜⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⢀⠜⠀⢀⢰⡟⠃
⠀⠀⠀⢠⡿⢀⣮⡼⠁⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠾⣿⣿⠃⠀⠐⠄⠛⠁⠀⠀⠀⣀⠄⠀⢀⠄
⠀⠀⠀⢸⢃⣾⣿⣧⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⠁⢥⡀⠀⠀⠀⢀⠀⠈⣛⣷⣾⣯⡄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣼⣿⣴⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠋⠀⠀⠀⠀⢈⡿⠿⢿⣿⣿⡛⠓⠂
⠀⠀⠀⢸⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠐⢉⠻⠶⠇⠉⠋⠉⠉⠀⠀⠀⠒⠈
⠀⠀⠀⠸⢻⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⢐⠁⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢸⡆⠀⡇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⢀⢿⡏⠀⠀⠀⠀⠀⠀⠐⢄⡀⠠⡠⣄⢠⡄⡟⠀⠀⠀⠀⠀⠀⢀⠠⢤⢀⣀⠀⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⢀⠀⠀⠙⢷⣴⣿⡛⠓⠃⠀⠀⠀⠀⠀⠀⠹⠷⢮⡁⠠⠙⠿⠟⢢⣄
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⡁⣦⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⢀⡇⠀⠈⡁⠄
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⢠⣾⣷⡇⠀⠀⠀⠀⠀⠀⢢⡜⢿⣿⣾⡉⠐⢛⡐⠛
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠈⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠠⣈⠀⠄⠀⠗⢁⠄⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠶⢄⠀⠀⠀⠀⠀⢀⣴⠰⢦
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⢀⡀⡠⠐⣲⣷⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⠭⠁
⠀⠀⠀⠀⡴⠛⣻⣟⣿⣡⢾⠏⠚⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣷⠏⣯⡉⠀⠀⣠⣶⡒⣂⣤⣤⡀
⠀⠀⠀⠀⠀⢸⠟⠹⠓⡹⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠁⡙⢀⠀⠀⠋⠘⢷⠈⠋⠗
⠀⠀⠀⠀⢀⠎⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⡀⠀⠑⠠⡀
⠀⠀⠀⡠⠃⠀⡠⠋⠀⠤⠀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡈⠑⠀⡀
⠀⠰⠞⢁⣴⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠗
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⢿⠿⡿⡿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡯⣀⠀⠀⠀⠀⠀⠀⡼⠷⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠘⢷⣦⡀⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⣿⣿⣿⣾⣿⣇⠘⣤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠴⣻⢿⢿⠿⡟⡏⠁⠉⠛⠛⠛⠛⠻⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⣯⡟⡼⠈⡀⠐⠂⠀⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠠⠤⠐⠈⠀⠀⠀⠀⠀⠀⢘⡟⠠⠁⢸⣧⣀⡰⠼⠟⠓⠓⠓⠛⠃
⠀⠀⠀⣠⣦⣾⣷⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠀⠀⢀⡉⠉⠀⠀⣸⠀⠀⠀⠀⣠
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡆⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⢀⣤⣿⣷⣤⣥⠶⠛⠛⠛⠛⠛⢣
⠀⠀⠸⠿⠁⠋⠛⡉⣿⢹⠀⢸⠛⠁⠀⠀⢠⣴⣿⠏⠀⠀⠀⢼⣿⢹⡥⡍⠀⠀⣹⣀⣀⣀⣀⣄⡇
⠀⠀⣤⠀⣦⣄⣦⣷⣿⢸⠀⠈⣿⡿⣷⣚⣻⣿⡏⠀⠀⠀⠀⠘⢿⣾⣇⣿⠷⠾⠟⠛⠛⠛⠛⠛⢇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⢸⠀⠈⣟⠀⠀⢸⠇⠀⠀⠀⠀⠀⣤⣤⢸⡇⢹⠉⣿⡆⠀⠀⠀⠀⢀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡆⣇⡀⠘⠀⠀⢸⣳⣄⠀⠀⠀⠀⠘⢿⣾⣧⣼⣶⣿⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠘⡿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⣀⣿⣿⣷⣦⡄⠀⠀⠀⠹⠏⠻⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠙⠉⠉⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⣤⣄⣀⠀⣤⣤⣤⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⡈⣹⣿⣿⠈⠹⣿⣄⣿⡏⣿⢳⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣭⡿⣓⢣⣀⡈⠟⣽⣯⣿⡃⠀⠸⠟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣡⡾⠶⡀⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⣩⡽⠗⠀⣿⡜⣿⣧⢸⣿⠟⢠⣿⢦⡀⠀⠴⢋⣆
⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⡴⢛⣥⡾⢻⣷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣻⡟⠁⣀⡀⠸⣿⣮⡙⢂⠃⣠⣿⡇⠀⣿⡶⡀⢠⣿
⠀⠀⠀⠀⠀⠀⣴⠻⣿⠀⠈⣰⡿⢋⣠⣾⣿⣆⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⠟⠃⢶⣄⢮⣅⡀⠈⢻⣿⠸⢰⣿⣿⠀⢰⣿⠁⠈⢻⣯⡂
⠀⠀⠀⢀⣴⡇⣿⣄⠘⠀⣰⠟⠠⠟⠉⢸⣿⡟⣆⠃⠀⠀⠀⠀⠀⠀⢠⡆⣸⣿⡜⢷⣦⡈⢿⣦⡙⢻⣧⢸⣿⠀⣯⣿⡟⢀⣿⡿⠀⢀⣼⡿
⠀⠀⢀⣿⣿⠁⢻⣿⡄⠀⣼⠀⠀⢶⡄⣾⢸⡇⣿⡦⠄⠀⠀⠀⠀⣾⣿⣿⣿⣿⠁⠀⢹⡿⠀⣿⡧⣠⣿⣹⣿⣾⣿⠟⣶⣿⢟⣠⣶⣿⣿⠃
⠀⠀⠸⣿⣿⡀⢸⣿⣿⢘⡿⢀⠀⠀⠀⢸⡘⣷⣿⡁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣶⣶⠞⠓⠊⠉⠉⠉⠉⠉⠀⢹⣁⣾⠟⣷⣿⣿⡿⠟⠁
⠀⠀⠀⣿⣿⡇⢸⣧⣿⠘⠇⢮⠀⠀⢀⠘⢿⣜⢿⡇⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣸⠟⠋⠁
⠀⠀⠰⣿⣿⣿⠈⠏⢿⡇⠀⠹⣇⡆⣿⠀⠀⢉⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢶⣾⢿⡿⣶⣟⠃⠈⢶⠃
⠀⠀⠀⢿⣿⣿⡆⠀⠈⢿⣶⣤⠈⠓⢿⠀⠀⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣣⣖⠉⠛⣇⡹⢦⣄
⠀⠀⠀⠀⠻⣿⣧⠀⠀⠈⠙⠛⠀⠀⠈⢷⣶⣾⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⣠⣴⣧⡎⠀⠈⠃⢰⡇⢩⢆⣼⠧
⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⣀⣼⣛⡿⡇⠻⣦⡀⠀⠘⠷⢼⣬⣟⣤⣾⣆
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣦⣤⠀⠀⠀⢀⡀⠀⣤⠀⢀⣀⠀⠀⠀⠀⢀⠤⢾⡷⢻⡏⠀⣿⡀⠈⠙⢶⣤⡀⠀⠀⠉⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⢢⣾⠀⢀⡏⣴⡟⠁⢠⣾⠀⢠⡟⠀⣸⡟⢸⣧⠀⠸⣷⡀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠈⣿⡄⠀⣿⣿⠁⢀⣿⣿⠀⣹⡇⠀⣿⠁⠈⢿⣆⠀⢹⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠘⣿⣆⢿⡇⠀⣾⠿⣿⠀⣿⡇⠀⣿⠀⠀⠀⠙⢦⡀⠙⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠻⢿⣦⡈⣿⡘⣇⠀⣿⠀⣿⠀⠹⣇⠀⠙⣧⣠⣶⣶⣄⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⢸⣧⣤⡉⢻⡟⠐⠿⢄⣹⡀⠘⢧⣄⠙⠧⣄⠈⢻⣤⣝⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⢀⡀⠈⣙⠛⠏⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠈⠀⠀⠈⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠚⠛⢷⣛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣉⠛⠘⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣘⣷⣤⣧⣾⡻⠋⠉⠙⠛⠲⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣣⠅⠼⠘⠞⠒⠝⠸⠞⢀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣬⣭⣻⣿⣿⠵⠾⠿⠿⠒⠶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⢾⡿⢛⡿⣯⣛⠟⠻⣶⣤⡈⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠁⠞⣵⣿⡆⣇⠙⣦⡀⠈⠙⠆⠙⠅⠀⠀⠀⠀⣠⣼⣷⡛⣚⠷⠦⠀⢀⣳⣠⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⡄⣤⢻⣿⡄⠹⢶⣽⣿⣄⠀⠀⠀⠐⡀⠀⠐⣣⢞⢿⣷⣬⠈⢳⢤⣴⠞⣻⣿⠛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢡⠻⠘⣿⡇⠘⠀⢹⡷⠙⠐⠆⠀⠀⠀⠀⣸⠧⠈⠈⣿⣟⢷⡀⢸⡏⢺⡏⢌⠓⣆⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⠀⠀⢠⡀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠸⣿⡁⠀⠀⠀⠁⠂⠰⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠦⣄⡀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠠⡀⠀⠈⠙⢦⣀⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠤⠬⠯⠿⠿⠤⠴⠖⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣆⣤⡀⠀⠀⠹⣿⡄⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢈⣉⡀⣠⣤⠴⠚⠋⠀⠀⢠⢴⠖⠈⠀⠀⠤⠐⠂⠙⢿⣯⢦⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣤⡤⠶⠶⠶⣶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠈⠐⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣯⣷⡀⠀⠈⠳
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠤⢤⠤⣤⡤⠂⢀⡤⠖⠲⠿⠟⠛⠋⢀⡾⠀⠀⠀⠀⠀⢀⣤⠖⠒⠈⠁⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⢞⣋⣩⣥⣤⡤⣶⢶⡶⠶⢶⣦⣤⣄⣀⣀⣀⣠⣤⡤⡶⠚⠋⠀⠀⣀⣀⠤⠚⢉⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠠⠤⠤⠤⠤⠴⠶⠖⠒⠒⠂⠉⠉⠉⠉⠀⣀⣀⡤⠨⠀⠀⠴⠒⠊⠁⢀⡀⠄⠚⠁⠈⠀⠀⢀⣜⠛⣋⣲⣆⣤⡤⠟⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢉⣉⠥⠀⠀⠀⠀⣀⠠⠄⠀⠲⠉⠀⠀⠀⠀⠀⣠⡖⣥⣦⡾⢸⡞⠈⢙⡓⠢⠄⠀⠀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠒⠒⠒⠒⠚⠉⠀⣠⠴⠒⠋⠉⠀⠐⠂⠀⠠⠂⠀⢀⠤⠐⠛⠱⠋⠽⠇⠀⠙⠋⠙⠛⠑⠃⠀⠀⠀⠀⠈⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⣀⠀⢠⠀⠐⠀⠠⠀⢀⠀⠀⠀⠄⠀⣤⠀⠀⠄⠠⠀⠀⠀⠠⡄⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠂
⠀⠀⠀⠀⠀⢀⠈⠀⢨⠀⠀⡀⠀⠀⠘⠂⠀⠀⡂⢰⣠⡼⠃⠀⠄⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠶
⠀⠀⠀⠀⣦⣈⠀⠀⠀⠀⠀⠀⢈⡅⠀⠀⠀⠸⠅⠸⠎⠰⠿⠀⠀⠀⠈⠀⡀⠉⠀⠐⠂⠀⠀⠀⠀⡀⠀⢠⡀⠀⢀
⠀⠀⠀⣀⠐⠪⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠠⠀⣤⠚⣿⠆⠀⠈⠁⠀⠀⠐⠀⠀⠰⠂⠀⠀⠀⠀⠠⠂⠠
⠀⠀⠀⠀⠂⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠽⠆⠋⢀⠀⠁⠚⠀⠀⠀⢀⣀⠀⡄⠚⠀⠀⠀⢀⡀⣀⠈⠂⠊
⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⢤⠄⠀⠀⠠⣶⣦⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠂
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄⠀⠈⠀⠀⠄⠀⠘⣿⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠠⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠐⠂
⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣼⣿⣿⣶⣷⣶⣶⣶⣿⣿⣆⠀⠀⠘⠳⠂
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⣀⠀⠀⢿⣿⠻⢿⠿⡟⢻⡿⣟⣿⡏⠀⠀⠀⠀⠀⠀⢀⠛⠠
⠀⠀⠀⠂⠀⠒⣠⡀⠀⠀⠀⠈⠀⠛⠃⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⡀⠀⠤⡄⠀⠀⠀⠖⠒⠂
⠀⠀⠀⠤⡀⠀⠐⣿⣾⣋⠀⠀⠂⠀⠀⠒⡴⠂⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠗⠢⠄⠐⢒
⠀⠀⠀⠀⠀⠀⠄⠇⠀⠤⠅⠀⠀⠐⠚⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠠⠀⠀⠒⠀⡠⢆⠀⡀⠀⠀⡐
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠒⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⢤⡂⠀⠣⠰⠂⠁⠀⠰⠉⠆
⠀⠀⠀⠀⠀⠰⡄⠀⠀⣠⣀⣸⡄⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣉⠙⠂⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⢄⠉⠁⢠⠴⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠈⠁⠀⠁⠁⠀⠉
⠀⠀⠀⠀⠠⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⣀⠀⠀⠄⠀⠀⠀⠀⠠⡢⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠈⠑⠀⠆⠄⠀⣀⠀⣠⠀⠀⠀⠠⠀⣀⠀⣶

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠈⠉⠉⠉⠉⠲⢠⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣀⢀⠤⣀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⢸⣦⣀⡀⠀⠀⡆⢻⡿⢂⡙⠃⠀⠀⢀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⡏⠉⠈⠻⢆⡀⠈⠁⠈⠋⠉⠀⠀⢠⡞⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⡇⢀⣀⠠⢾⡃⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⠀⢨⡿⠦⢀
⠀⠀⠀⠀⠀⢃⣘⡏⠂⠴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⢻⣄⣄⡴⠀⠀⠀⢀⡠⠶⠦⠶⢾⣟⠁⠀⠀⠀⠀⠀⠙⢿⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⢀⡴⠟⠡⡄⠀⠀⠀⠀⠘⠷⢦⡀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢰⠃⠀⠀⠀⠈⠑⢦⣀⣤⠆⠀⠀⠈⠐⠠⣀⠀⠀⠀⠀⠀⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡊⠀⠀⡀⠀⣠⠀⠀⢹⡄⠀⠀⢀⡀⠀⠀⠈⠓⣦⣀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢿⠄⠀⠋⠀⠀⠰⣶⡏⠀⠀⢤⠋⠁⠀⢀⣀⣠⠄⠉⠛⢦⣀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⠀⢸⡀⠀⠀⠀⠀⠑⢀⠀⠀⣦⡀⠀⣠⠀⠁⠀⠀⠀⠀⠈⠳⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠉⠁⠀⠀⠀⠀⠈⢷⠸⡁⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣹⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠱⣄⠀⠀⠷⠴⣦⣄⡀⠀⠀⣀⠛⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣀⡷⠠⡀⠀⠀⠈⠉⠛⠛⢧⣄⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠀⠈⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠀⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈⠐⣿⠀⠀⠀⠙⣦⡀⠀⠀⠀⣤⠀⠀⠠⠿⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠙⠁⢄⣀⡠⠦⡈⠳⣤⠄⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠂⣀⠁⠀⠀⡀⠀⠀⠀⡀⠈⠀⢄⠈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⠐⣄⡀⠀⠄⠀⠀⠀⠈⠙⠲⠢⣬⠒⣤⠀⠀⠁⠒⠀⠈⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠒⢴⣶⡖⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣤⣤⣤⣀⣀⣀⣀⣀⣤⣶⣿⣿⣟⣛⣿⣷⣦⣤⣤⡀⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⢻⡜⠄⠈⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠈⠁⠀⠙⣿⣯⠀⠀⠀⠀⠀⠀⠈⠳⣬⣙⠓⠀⠀⠀⠀⠀⠀⠀⢱⠈⠐⢿⠁⠀⠉⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠙⠲⢤⣄⡒⠢⣤⣤⠼⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣛⣡⣤⣤⣄⠀⠶⠶⠶⠛⠀⠸⣆⠀⠀⠈⠙⠳⣤⣀⣄⡀⠀⠀⣤⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠸⡅⠉⡟⠛⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢿⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠘⢇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠘⡀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢃⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠩⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐





⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡴⠂⠀⠀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡀⠀⠀⠲⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠐⢿⣥⣼⡆⠀⠀⠀⢿⣌⢲⣍⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢈⣴⠋⣼⠇⠀⠀⠀⣾⣤⣽⠗
⠀⠀⠀⠀⠀⠀⠸⡿⠰⣇⠀⢀⠤⠶⣻⣍⣀⢻⡄⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⣼⢃⣀⣽⡳⠦⠄⠀⢀⡷⢸⡿
⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣤⠤⠀⠀⠡⢿⣿⣄⣡⡄⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⣤⣁⣼⣿⠧⠀⠀⠠⢤⣼⣷⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣷⡶⡀⠀⠈⠁⣻⣿⣿⣿⢀⡱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠼⡁⢸⣿⣿⣿⡃⠈⠀⠀⡰⣶⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣡⡀⠀⢈⣴⡿⣿⣿⠀⢷⣆⡛⣁⠤⢴⠢⢄⡙⣃⣶⠃⢸⣿⡿⣷⣌⠀⠀⡠⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢮⡳⢄⠀⠰⠿⣿⢀⢏⢿⡿⣆⢲⣩⣱⢋⡼⣿⢏⠇⢸⡿⠳⠀⢀⢔⡫⠞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣌⢙⣀⣰⡇⡀⣸⢸⣸⣇⠼⣳⠍⢴⡻⢄⣿⣸⢸⠀⡀⣧⣣⢘⢉⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠻⠿⡿⢷⣆⢻⣦⠑⢻⡟⢫⡞⣬⡛⣭⠛⣡⣾⠋⣶⠿⡿⠿⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠦⣹⣿⣷⣿⣳⣦⣙⠛⠉⠘⢛⣡⣴⣻⣿⣿⣿⡡⠖⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠇⠀⢨⣾⣿⣿⢿⡜⣩⣽⣿⣿⣭⡛⣼⢿⣿⣿⣮⠀⠀⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⠗⢤⠉⣿⣿⢫⣼⣷⣿⠻⣿⡿⢻⣷⣿⣬⢻⣿⡏⢡⠴⢷⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣈⡷⠚⠿⣿⣿⣿⣹⣟⣻⣿⣿⣛⣿⣹⣿⣿⡿⠟⠲⣏⡈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⢻⣿⣿⣿⣿⠻⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠋



⣄⣀⣀⣤⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤
⣟⣻⣛⣛⣟⣟⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠿⣿⣿⠁⠀⠀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⢀⣀⠀⠀⠀⢸⣿⣆⣀⣀⡴⠲⢤⣀⠀⠉⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠉⠁⢀⣀⣀⣤⡤⠤⠶⠖⠚⠛⠋⠉⠙⢦⣀⡀⠀⣿⣟⠉⠁⠀⠀⠀⠈⠙⠓⠦⣄⣀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⡀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢲⠋⠀⠀⠀⠀⠀⢠⡞⠛⢷⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡆⠀⠀⠀⢠⡞⠉⢳⡄⠀⠀⠀⠀⠸⣇⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡆⠀⠀⠀⠀⠀⠘⠧⡤⠞⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣇⠀⠀⠀⠈⠳⠤⠞⠁⠀⠀⠀⠀⠀⢹⡇⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⠀⢻⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣄⡀⠀⠀⢸⡄⠘⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⡞⠀⣸⣿⣧⣀
⠀⠀⠀⢀⣠⣶⡿⠟⠉⢹⣿⡀⠸⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⠀⠈⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡾⠁⣰⣿⠋⠙⠿⣷⣦⣀
⠀⣠⣶⡿⠟⠉⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⣾⣿⡿⠋⣿⣿⠁⢀⣀⠈⢿⣿⣇⠀⢿⠀⢹⣿⠋⣀⡀⠙⣿⣿⡇⠀⠀⣼⠁⢠⣿⠏⠀⠀⠀⠈⠙⠿⣷⣦⡀
⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⠀⢸⣿⡿⠁⠰⢟⡋⠀⣭⣭⠀⢰⣿⣿⠀⢸⡄⠘⣿⣷⣭⡃⠀⣿⣿⡇⠀⣸⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡇⠈⣿⣿⣆⠈⠉⠁⢠⣿⣇⣠⣾⣿⡟⠀⠈⣇⠀⣿⣿⣀⣠⣾⣾⡿⠀⣰⠃⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿
⣿⡏⠙⠻⢿⣶⣄⡀⠀⠀⠈⣿⡇⠀⣧⠀⠘⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⢻⠀⢸⣿⣿⣿⡿⠋⠀⣰⠏⢀⣾⡟⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⣿
⣿⡇⠀⠀⠀⠉⠻⢿⣶⣄⡀⢻⣿⠀⢹⡀⠀⠀⠉⠻⠿⢿⣿⠿⠿⠛⠁⠀⠀⠀⠀⢸⡆⠘⣿⡏⠁⠀⠀⢠⠏⠀⣾⡿⠁⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣷⠀⠀⢠⡏⠀⣼⡿⠁⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⢠⡴⠲⣤⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿⡀⢀⡞⠀⣼⣿⣷⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⡀⠀⠀⠀⠀⠀⠸⣄⣀⡼⠀⠀⠀⠀⠀⠀⣸⡆⠈⣿⣇⡾⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢿⣿⠁⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠛⢦⡀⣀⣀⣠⣤⠤⠴⠶⠒⠚⠛⠉⠁⠀⢀⣀⣾⡿⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⠉⠉⢀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⡿⠛⠋⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣶⣶⣿⠿⠿⠿⠛⠛⠛⠉⠙⠿⠟⠉⠙⠻⢿⣷⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠙⢿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢄⡀⡰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠀⠀⠀⢰⡷⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣾⣏⣤⡀⢀⣄⠀⣀⡠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢓⢠⡅⡥⠛⠋⠻⠿⠇⠛⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⣰⣣⡀⠐⠀⢀⣦⡙⠿⠃⢠⡿⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠐⠂⣠⣦⣦⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠈⠟⠁⠀⡼⣷⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⡁⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠛⠉⠁
⠀⠀⠀⠀⠀⠀⣿⣟⡿⣫⠀⠀⠀⠀⠀⣽⣿⣿⡇⠀⠀⠀⠀⠀⣾⣶⣦⢠⠀⠘⠿⠀⢠⣄⣴⣶⣦⠀⠀⢤⣰
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⢐⣾⡆⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠘⠺⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⢀⠀⠀⢀⠀⠀⠙⠋⠀⠀⠈⠉⠀⠀⠈⠻⠃⠀⠁⠀⠀⠀⠀⠀⡹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⠛⠁⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠆
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠸
⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⣀⡀⠀⣀⡀⠀⣰⣿⡄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢠⣄⡀⠀⠀⢀⣠⣼⣧⡀
⠀⠀⠀⠀⠀⢰⣿⣿⠟⠛⠻⣧⢠⡿⠛⠛⢿⣿⣷⠀⠀⠘⠔⢠⡄⠀⢀⡴⠻⣿⠟⢷⣄⠀⠀⡤⠸⠎⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡏⠠⠀⠀⠀⣿⣿⠀⠀⠚⠚⢓⣰⠆⢸⣿⠀⡁⠀⢘⣿⠀⠖⡐⠃⠚⠀⠀⠀⢰⡿⢋⣙⠻⠋⣩⡉⢿⣿⣿⣧⠀⠀⣀⣀⣄
⠀⠀⠀⠀⠀⢸⣿⣷⣌⣉⣠⣾⣿⣤⣈⣉⣴⣿⣟⠀⠀⠀⠘⢚⡡⡄⠀⠻⣶⣿⣷⡿⠋⢀⠜⡑⠛⠀⣿⣿⣷⣾⡇⠘⠛⢠⡄⠛⠁⡈⣿⣿⣿⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢸⣿⣿⡏⠉⠁⠈⠋⠀⠉⠉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡏⠃⠀⠀⠀⠀⠈⠁⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢸⣿⡿⠁⣠⣾⣿⣿⣿⣦⡀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠐⠁⠓⠏⠂⠙⠁⠁⠛⠀⠀⠀⠀⠀⠙⠯⣿⣽⣆⠀⠀⠀⠀⠀⣠⣿⣿⣯⣿⡽⠟⠁
⠀⠀⠀⠀⠀⠘⠟⠁⣼⣿⣿⣿⣿⣿⣿⣿⠆⠙⠏⠀⣤⣄⣀⠄⠀⡀⡀⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⠀⠙⠶⣤⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠿⡿⠛⠁⠀⠀⠰⠒⢲⡄⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣴⠒⠲⠆⠀⠀⠀⠀⠿⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠦⠀⠀⠀⢠⣶⠞⢁⡇⠁⠀⠀⠀⠀⠂⠀⠀⠂⠁⠀⠀⠀⠀⠀⠁⣡⠘⠦⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⡟⠓⠀⠀⣻⣿⣿⣿⣿⠃
⠀⠀⣼⣿⣿⡟⠉⠉⢻⣿⣿⣷⣀⠀⣀⣀⣤⣤⣤⣤⣤⣦⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⠀⠀⣺⣿⣿⣿⣿⣮
⠀⢼⣿⣿⡿⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓⠐⣿⣿⣿⣿⣿⣿⠿
⠀⣿⣿⣿⡇⠀⠀⠀⠀⡔⢉⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⡇⠀⠀⠀⢰⣶⢨⠀⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡇⣌⣹⣿⠉⠈⠀⠉⠛⠁
⢰⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿
⢘⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢿⣿⣿⣿⣿⡿⠁
⢸⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⢃⣴⣾⣿⣿⣿⡿
⠘⢿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡉⠙⠻⢿⣿⣿⣾⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⡿⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇⠙⠻⣿⣿⣷⣦⣤⣤⡀
⠀⠀⠸⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠘⢿⡿⠛⠛⠿⠛
⠀⠀⠀⠙⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀
⠀⠀⠀⠀⢣⣽⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠛⠁⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡷
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣻⣿⣏⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣀⣀⣀⣀
⣿⣿⣋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⠈⠙⠳⢆⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⣦⡤⣤⣼⡀⠉⠙⢿⣶⣤⡈⢷⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣍⣿⣿⣿⣷⣄⠀⠀⢹⣿⣇⠈⠐⠄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⣩⠟⣱⣿⣥⣤⣨⣿⣿⠒⠀⢀⡈⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⣿⠁⠿⢿⣿⣿⣿⠿⠏⠀⠀⠀⠁⣠⠈⣢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⣾⣿⣿⡀⠀⠀⠀⠠⣾⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⣿⣿⣿⣿⣿⡟⠻⠿⡮⠀⠒⠂⠐⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠈⣴⣿⣦⣤⣙⣧⠀⠀⠻⠄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢠⣿⣿⣿⠿⣿⣿⣷⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣾⣿⣿⠁⢀⣾⣿⠛⢿⣴⣾⠿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣿⠿⠿⣿⣿⣷⣾⢿⠍⠀⠉⠉⠀⠀⢸⣿⠛⠳⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠅⠀⠀⠀⠠⠀⠀⠀⠀⠈⢫⡀⠀⠀⠻⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢁⡒⠁⠀⢀⢠⣤⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠎⠁⡀⠀⠀⠀⣤⣤⢟⠛⠋⠀⠀⣴⣿⣿⠯⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣠⡀⢠⠄⢪⣼⣿⣿⣯⣴⣿⣿⡿⠛⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣟⡛⠴⢾⣾⣿⣿⣿⣿⠟⠋⠁⠀⠀⠰⣶⡌⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠾⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣰⣔⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⡤⣴⣿⣿⣯⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣥⣶⣿⣯⣷⣾⣿⢻⡟⣹⡇⡄⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⣦⣄⡀⠀⠸⡁⠤⡤⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠐⠸⠿⠿⠉⠀⢀⠜⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠠⠀⠀⠀⠀⠀⠀⢸⠦⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⠀⠈⡄⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠉⠈⠓⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⢈⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢀⡀⢰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠈⠁⠐⡁⢀⣀⣄⣀⠀⠀⠀⠀⠔⠠⠊⠀⠈⠴⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⠁⠀⡀⠄⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⡁⡀⠀⠁⠀⠁⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠴⠀⠀⠀⢠⠄⢴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠒⠈⠁⠂⠁⠀⠜⠁⠀⠀⠈⠂⣴⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢸⣣⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠉⠉⠁⠘⠂⠲⣄⠂⠠⠀⠀⠀⠀⠀⠤⢠⣔⣈⠠⠐⠁⠀⠀⠀⠀⢀⠀⡩
⠀⠀⠀⠀⠀⣀⠤⠔⡊⠀⣖⠋⠀⠐⠀⠀⠐⠒⠁⠙⠚⠧⢠⡴⠄⠚⠑⠈⠀⠀⠀⠀⠀⠀⡀⠀⠰⡶⣡⠔⠁
⠀⠀⡠⠆⠉⣀⠐⡏⠑⠀⠀⠀⠀⠈⠀⢐⡒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣠⠾⠊⠁
⠀⠀⠁⠀⠀⠉⠀⠀⠐⠒⠀⠈⠁⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢎⣴⠏⠀⠰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠔⢀⡀⣀⢴⢾⣠⣤⡔⠀⠀⠀⠀⠀⣀⡀⢀⠄⢀⡠⠒⠚⠋⡏⠚⣫⡤⠀⡜⠉⠉⠺⢆
⠀⠀⠀⠀⠀⣀⡴⢒⠅⠊⠀⠈⠀⠀⠀⠀⠉⠘⠀⠀⠚⠿⠛⠁⠈⠁⠰⣄⣀⠴⠔⠋⠀⡄⠘⢤⠿⣱⠢⡀⢱⡦⡆
⠀⠀⠀⡠⢚⠥⠊⡡⠀⠀⣀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⠀⠈⠙⠛⠃
⢀⢔⡠⢒⣡⠔⠁⢀⠴⠊
⠳⠈⣴⠋⣀⠤⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠛⡏⢩⡀⠖⠂⠄⠀⠀⠀⠠⠂⠈⠛⢩⣈⡉⠉⠹⢒⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠻⠄⠾⢳⡒⠈⠁⢀⣄⠀⠀⢠⠖⠀⠀⠲⡀⠀⠀⠙⠈⠉⠋⠀⠉⢠⡄⢀
⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⡀⢦⠀⠀⢤⣴⣶⠾⡿⣿⢯⣍⡉⠙⠉⠁⠘⠁⠀⠉⣀⡤⠷⠛⠟⢿⢿⣾⣁⠀⠀⠁⠶⣀⢤⣴⣆
⠀⠀⠀⠀⠀⠘⣧⠲⠡⠐⠲⠀⠀⠀⣾⠛⡤⣲⣤⣤⠾⢭⡛⣦⠀⠀⠀⠀⠀⣴⠋⠡⣐⣬⣤⠖⢾⡜⢻⣧⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣿⡄⡌⣿⣿⣿⣿⣤⣌⠁⢴⡆⠀⠀⢀⢺⡍⢌⣱⣿⣿⣿⣦⣴⠀⡄⢾⡇⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡆⠀⣿⣽⣿⣿⣿⡟⣀⣰⠇⠀⠀⠘⢻⣅⠐⢻⣿⣿⣿⣿⣟⠀⠀⣻⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠻⣷⣈⠟⣛⣟⠋⠐⣥⡿⠄⠀⠀⠀⠀⣿⣆⠑⠙⢛⣛⡉⠶⢀⣼⡯⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠋⠙⠺⠿⣾⠶⢿⠁⠀⠀⠀⠀⠀⠀⠋⠙⡳⠦⣤⣈⣤⣴⠿⠏⠀⠀⣠⣾⡛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⣤⡀⠀⠀⠀⠀⠈⠉⠈⠁⠀⢀⣠⡾⢿⡉⢷⡦⢟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣴⠋⢙⣽⢿⣾⡷⣶⠖⠄⠤⠂⠒⠊⣿⣿⣿⣿⡟⠐⠀⠴⠤⠂⠠⠶⣴⠾⣿⠛⣷⡈⡿⠚⣦⣈⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢡⡤⢾⠃⣼⠃⣾⠉⢠⣤⣤⣤⣤⣴⢸⣿⣿⣿⠇⣶⣶⠧⡴⣶⢶⠾⠿⡄⠹⢧⣾⡁⢻⡆⣼⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠶⣇⣰⣿⠃⢠⡟⠙⠶⠿⣀⡞⠀⠀⢀⠀⢠⡆⣿⣿⣿⢀⣇⡴⠀⠀⠀⠀⢐⣲⣿⣧⡀⠻⠓⢶⣄⢸⠗⢻⣄⣴⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣞⠈⢩⡄⠛⣇⣼⠇⠀⠃⣿⡌⠁⣬⠘⢛⣷⠹⣿⡏⣈⣿⠉⠉⠖⠀⠀⢴⠄⢼⠶⠋⢳⡦⣼⠃⢻⣦⣼⠛⠿⡀
⠀⠀⠀⠀⠀⠀⠿⠛⠛⡤⠾⠇⣠⡇⠀⣼⠛⠶⠟⠃⠀⠛⠢⠈⠉⡀⣿⡇⢨⠍⢀⠀⣀⠠⣁⣲⣄⢂⣹⣷⣯⠀⢹⠇⣾⠋⠈⠡
⠀⠀⠀⠀⠀⣠⠶⣤⠾⣇⢀⣿⠉⢡⠚⠛⠶⢃⣀⡈⣁⣤⣀⣈⣴⣧⢸⢃⡾⠿⠛⠚⠛⠓⠈⠉⠉⠉⢉⠙⢠⡷⢿⡄⠸
⠀⠀⠀⠀⠀⠋⣀⠏⠀⢸⠃⣙⠒⠛⣤⠾⠃⠛⠋⠃⠉⠉⢡⠉⢀⠈⡄⢳⠀⢀⠀⣄⠀⠀⠶⢮⠐⠷⠀⠰⠛⣷⡀⠀⠀⠀⠀⠀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⡴⢆⠸⡀⠟⠀⢂⡙⠀⠀⡈⠁⠁⠚⠁⠀⠀⢀⣠⠀⢤⠀⠴⠒⠖⠐⠛⣿⠙⢢⣀⠄⠀⣆⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⣄⠀⠀⢀⣴⣀⡤⢠⠶⡴⠂⠐⠤⠖⡤⠁⠞⡡⠛⠚⠁⠃⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⡟⡀⠀⠙⣿⣦⣸⣿⡇
⠀⠀⠀⠀⠀⠀⢤⣄⢸⣿⣷⣷⠈⠉⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡅⠀⠀⠘⠏⠁⠀⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢿⡇⠠⢠⣤⣤⡤⠖⠖⠤⠨⠁⠌⠀⠀⢀⠈⠠⠀⣅⠀⠀⠀⢁⣤⡀⠀⠀⠤⠀⢵⡟⡇⠀⠀⠀⠀⠐⠤⠁
⠀⠀⠀⠀⠀⠀⠀⡤⠟⠀⢺⣍⣥⣠⡄⣤⡤⣤⢤⡆⣶⡆⣶⠠⣦⢠⡤⣴⠤⣤⢤⡤⣴⢤⣦⢰⡦⣶⢠⡆⣴⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠤⢀⠴⠀⣀⢂⠀⡀⣀⠀⢀⠀⡀⢀⠀⠀⡀⠀⢀⠀⡀⢂⠀⡀⡀⢀⡀⡀⢈⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡞⠙⠳⠆⠐⠤⠠⠀⠀⠈⠡⠈⡄⠀⠀⠀⠀⠉⠀⠀⠆⠁⠈⠂⢠⠤⣈⠄⣠⣬⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡆⠃⠚⠐⠠⠂⡆⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠂⠉⠀⠁⠆⠰⠀⠆⢰⡇⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢦⠐⠆⠢⠠⠰⠢⠀⠄⠀⠆⠀⠀⠀⠀⠄⠀⠀⠠⡄⠀⠀⠀⠄⠠⠦⡼⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⢶⣰⣦⣴⣌⣦⣶⣖⣶⢰⡖⣤⢤⡄⣶⢤⣤⢤⡄⣤⠀⣤⣠⣀⡼⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡑⣌⠛⠃⠈⠛⠉⠙⠋⠉⠿⠋⠙⠟⠘⢿⡞⠃⢾⠖⢋⣿⠟⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠦⢙⠳⢮⣷⡔⡐⠆⠲⠶⣄⡤⠦⢤⣠⣀⣤⣶⡄⢊⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠭⠁⣈⡒⣑⢚⣒⣚⣃⠋⠍⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⠚⠀⠀⠀⠀⠐⠓⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠤

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣭⣭⣯⣭⣽⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦⣶⣶⣶⣿⣿⣿⣦⣬⣽⣿⣶⣷⣼⣯⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⡴⢂⠐⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣧⠐⢀⣀⠀⠀⠙⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⢐⣭⠉⠸⠗⣈⣿⣶⠠⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣿⣷⣆⠙⠖⠀⣹⠃⠰⢁⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠉⠻⠿⣿⠦⠄⣀⡉⢠⡂⣤⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢗⣫⢹⠢⡬⣷⡍⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⣿⣿⡼⣿⡄⡺⢭⢭⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⢿⣿⢯⡽⣿⣿⡛⡯⣳⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠁⢬⣿⣿⣿⢗⠷⣿⠻⠖⣿⠽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣮⢁⣿⣼⣿⡝⣻⣯⣼⠿⡷⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠛⠿⣮⡟⣕⠌⢹⡿⣤⣤⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣜⣆⠻⣿⣝⣷⣾⣯⢿⢷⢋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡸⡤⠌⣿⡌⠉⢻⣬⣼⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡝⣲⣎⡻⣄⠀⢹⠙⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡰⠮⣆⠑⣛⣧⣀⡅⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢿⡴⢭⢥⣖⢺⡓⠟⢿⣿⣧⣀⠙⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⣄⢀⠀⠀⠀⢓⣸⠂⠀⢻⣿⣮⡳⣤⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠆⠀⠀⠀⠘⠁⠀⠀⠀⢿⡿⣟⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠒⠐⠉⠈⠛⠀⠀⠀⠀⠈⢻⡙⣯⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡄⠀⠀⠀⠀⠀⠁⠘⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠠⡀⠀⠀⢤⡀⠀⠂⢦⠀⠀⠚⠀⠀⠀⠀⢹⡆⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠄
⠀⠀⠀⠈⠢⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⡇⠀⠒⠰⠻⡧⢀⠀⠰⠖⠓⠀⢠⣀⠁⠈⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠒⠄⠀⠈⠁⢈⠴⠚⠁⠀⠀⣀⠁⠀⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠰⠁⠀⠀⠀⠉⠠⠐⠊⢀⠄⠈⠩⠵⠂⢀⣄⠄
⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⢸⣶⣆⠀⢆⠀⠀⠈⠀⠉⢁⣀⡤⠀⠒⢉⠜⡠⢀⠃⠐⠂
⠀⠐⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⢸⣿⡏⠀⠀⠁⠀⠀⠀⠀⠠⡶⠿⠷⠗⠚⢊⣴⣧⡀
⠀⠀⠀⠀⡁⠀⠀⡁⢘⠀⠀⠀⠀⣿⠀⠀⠀⠀⢺⣿⡇⠐⠀⠀⠀⢠⡆⠀⠀⠈⠓⠶⠠⠶⠻⠀⠙⠛⣠⠄
⠀⠀⠀⠀⡄⠀⠀⠁⢈⠀⠀⠀⠀⠸⠀⠀⣀⠀⠼⠛⠛⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠹⣿⣄⣀
⠀⠀⠀⠀⡀⠀⣀⠆⠀⢀⠤⠀⠄⠉⠉⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⢸⡇⢈⢀⡤⠐⠂⠀⠀⠁⠀⠈⢘⣿⠟⠋
⠀⠂⠄⠲⠀⠀⠀⠀⠐⠭⠿⠿⡖⢂⡀⠀⠀⠀⢠⣄⡀⠅⠀⠀⢨⣿⡏⢠⡠⠧⠖⠀⠀⠰⠏⡀⠀⠼⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠄⠩⢈⣀⣋⣉⣁⡄⡀⠀⠀⠈⣿⠧⠴⠓⣺⣉⠲⣄⠉⢳⠀⠀⠀⠤⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠟⣻⣥⢬⣴⣿⣽⣷⠂⠀⠀⢘⣿⣿⣅⠀⠀⠀⣨⠟⢀⣀⣤⡴⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠡⠖⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣅⠀⡀⠀⣰⠍⠩⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣦⣶⣶⣴⣦⣴⣿⣴⣷⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢴⣄
⠀⠀⠀⠀⠀⣀⠠⣄⣲⣾⠓⢛⣿⣷⢤⠀⠀⠀⠀⢤⣴⣺⠟⠁⠀⠣⡀⠀⣍⠉⢁⣶⣾⣫⠽⠛⠁⠀⠘⠈⠻⣶⣄⡀
⠀⠘⢿⠀⠀⠀⠀⠩⡍⠉⠀⠙⢻⣟⠀⠈⠀⠄⢀⠛⠛⢧⣤⣀⣀⠀⠙⢄⡇⣠⡾⠝⠋⠀⠀⠀⠀⠀⡇⠀⠀⠈⠻⣿⡷⡀
⠀⠀⠈⠳⣶⣤⣤⣄⣄⡀⠀⠀⠈⠉⠐⠤⣀⠀⡜⠑⠦⣀⠻⣏⣈⡩⠛⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⣀⣠⡤⠟⢳⣽⣄
⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣷⣦⣄⡀⠀⠀⠀⣽⠷⠶⠤⣌⣑⠨⣿⣁⡼⠃⠀⠀⠀⠀⠀⡀⠄⠠⠀⣺⣋⠉⠀⡁⠄⠊⠁⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣷⣦⣾⡏⠀⠀⠀⠀⠀⠈⠉⠻⣇⡀⠉⠀⠀⠀⠀⠀⠀⠀⣰⢻⡅⠀⢀⠠⠐⠀⠀⠀⠀⠘⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠇⠀⠀⠀⠀⠀⠀⠀⢀⠰⡎⠢⣄⠀⠀⠀⠀⠀⢠⠃⣟⡷⠈⠀⠀⠀⠀⠀⢀⠠⠐⣹⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣦⡀⠀⠀⠀⠀⠀⢐⠀⢹⡤⢊⣱⢦⣀⠀⢠⡆⠀⣧⣿⡀⠄⢀⠠⠄⠈⠀⠀⢠⣇⡈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣵⣄⠀⠀⠀⠀⢴⠀⠀⢡⡄⠈⢉⠙⢳⡟⠓⠶⠿⡿⢶⠚⠁⠂⠀⠀⠀⢠⣿⣿⠿⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣀⠠⠄⠾⣟⣿⡟⠙⠻⢷⡟⢿⡇⠠⣠⠚⠁⠀⠑⣄⠀⠀⠀⢀⣿⠟⠉⢠⣟⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠙⢿⢀⠎⢸⣷⣾⠇⠀⠀⠀⠈⠙⣿⠚⢀⠀⠀⠀⠀⠈⠠⡀⢀⡞⠁⠀⣰⡏⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢰⠎⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠉⠀⠒⠀⠂⣨⠾⣷⣄⣴⣿⣿⣶⣾⡟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣿⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⡆⠀⠐⢄⠀⢀⣠⣶⠟⠁⠀⠀⠙⢿⣧⢸⣿⠟⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣀⡟⠉⠓⣿⡟⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠈⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠹⣿⡿⠖⢩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⢀⣹⣀⣀⣼⡟⠀⠀⠀⢀⣀⣀⣤⣴⠿⣿⣦⣰⠟⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣷⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠋⠉⠛⠛⠁⠀⠀⠀⠋⠉⠉⠉⠉⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠰⣷⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⢻⣇⠀⠀⣿⠀⠀⣿⡇⠀⢠⡿⠀⢀⡖⠀⡴⢀⣴⠋⣠⡶⠋⠁⠀⢀⣀⣠⣤⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⢿⡇⠀⢸⣿⠀⢰⡟⠀⣸⣿⣧⠀⢸⡇⠀⣸⢁⣼⡷⠋⢠⠞⠁⠀⣠⣴⣾⠿⠛⠋⠉⠁⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠙⡇⠀⢸⡇⠀⢸⣿⠀⣸⠃⣼⠿⠿⣿⣆⠸⣧⠀⣿⣿⠏⠀⣰⠏⢀⣠⡾⠛⠉⠀⠀⣀⣀⣀⣀⣐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⢸⣇⠀⢸⡇⠀⢸⡏⢠⡟⣼⠃⠀⠀⠈⢿⣆⣿⠀⢿⡏⠀⢰⡏⢠⠟⠁⢀⣠⣴⠾⠛⠛⠉⠉⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡶⠀⠀⠀⠘⣿⠀⢸⢷⠀⣿⢁⡾⢠⠇⠀⢰⣆⠀⠈⣿⢻⡆⢸⡇⠀⣾⠇⠀⢀⣴⠟⠋⢀⣠⣤⡶⠶⠶⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⢀⣤⠀⠠⡀⢻⡆⠘⠀⠀⠀⠀⠀⠈⠀⢀⡿⠹⣆⠀⢸⡌⠃⢸⡇⠀⣿⠀⠀⡾⣃⣤⣶⠟⠋⠀⣀⠀⠤⠶⠀⣴⣤
⠀⠀⠀⠀⠀⠀⠀⡰⠀⣀⠈⠁⠀⠀⣷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠘⠆⠈⠃⠀⠘⠓⠀⠙⠀⠀⠈⢫⡟⢁⣤⠶⠛⠋⠀⣀⣀⠀⢻⣿
⠀⠀⠀⠀⠀⢀⡞⡠⢂⡈⢳⡄⢰⡇⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠋⠀⠀⠀⢀⡀⠸⣿
⠀⠀⠀⠀⢠⣿⣾⠠⡄⠳⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠆⠀⠀⢀⣀⠀⠙
⠀⠀⠀⢰⣿⣿⣿⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣈⡉
⠀⠀⠀⠈⠻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⣩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉
⠀⠀⠀⠀⠸⡄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⠀⣾⣿⣿
⠀⠀⠀⣧⠀⣷⣼⡟⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣠⣴⣾⣿⣿⣿⣿⣿⣷⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣛⣙⣟⣉⣛⣻⣋⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⢀⣀⡶⠚⠊⠁⢀⡀⣠⣀⡄⢀⣠⢠⣴⡟⠀⠀⠱⡄⠃⠃⠘⠀⢚⠏
⠀⠀⠀⠀⠀⠈⠷⠄⠀⠀⣘⡁⢼⡏⠀⠀⠀⢀⡟⠋⣻⣿⣗⣩⡾⢺⡿⢯⠀⠀⡀⢀⠈⠁⠠⣷⣿
⠀⠀⠀⠀⢰⠷⣦⣄⣄⣼⣿⢿⣭⡇⠀⠀⢢⣤⡄⢈⣽⠏⠾⠟⢠⣆⠀⠀⠀⣸⣿⠈⠁⠀⠀⠀⢇⠆
⠀⠀⠀⠀⢸⣄⠜⠛⢹⢦⠁⡆⠹⠊⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠉⠁⠀⠀⠈⢩⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠠⠈⠳⠄⡘⠸⡇⠀⢻⡗⠘⠇⠈⠀⠁⠈⣹⣿⣶⠀⠀⠀⠀⠀⠀⠀⠈⠰⠄⠀⠀⠀⠀⠃⠀⠂
⠀⠀⢀⠖⠀⠀⠀⠐⠄⢠⡅⠀⣸⣿⠀⠀⠤⠄⢂⡔⠁⠘⠻⠄⠀⠤⠖⠀⠄⠀⣠⠄⠀⠶⢄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡎⠛⠀⠃⠀⠀⡮⠘⠙⡿⠁⠀⠀⠀⠑⠀⠀⠀⠀⠀⢴⠋⠀⠀⠀⠀⠀⡸⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠾⠋⠱⢄⡀⠀⠀⠀⠀⠀⠠⢠⠆⠀⠀⠀⠣⠑⠀⠀⠀⠖⡞⠀⠀⠀⠀⠀⣠⠙⡐⡀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⡿⠇⠂⠀⠀⠀⠈⠷⠂⡀⠀⠀⠄⠀⠀⠀⢸⡇⠀⠀⠀⡠⠂⠈⠀⣸⣇⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢥⡶⢤⣉⡙⠷⡄⠀⠀⠀⠈⠉⠀⠀⠃⠀⠈⢹⠀⠀⢸⠚⣧⣀⣰⡿⡟⠻⣾⣯⣿⣷⠛⠋
⠀⠀⠀⠉⠀⢰⠀⢀⣀⠀⣂⠈⠙⡆⠙⡄⠀⠀⠀⠀⡀⠀⠀⡘⠧⠼⠆⠀⡺⠀⠌⢉⣿⡀⠛⠀⠈⣿⡼⡏⠇
⠀⠀⠀⠀⠀⠘⠁⠾⠋⠉⢳⣄⡀⢃⠌⢡⣦⣀⠀⠈⠁⠀⠀⣀⡀⠈⠀⠀⣟⡦⠶⢿⢏⣿⣶⣶⣾⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠠⡂⠀⢠⡿⠛⢁⠢⣻⡿⠉⠀⠀⠀⠀⢻⣿⠃⠔⣡⠞⠁⠀⠀⠀⣸⠛⠛⠛⠻⣿⣇⠹⣅
⠀⠀⠀⢀⢐⣤⠖⠂⠁⠀⠘⠃⢡⡌⠁⣻⠁⢰⣿⠇⠐⣿⠆⢻⠴⠞⠉⠀⠀⡉⢩⣿⣆⣀⠀⠀⠀⠈⣿⣶⡀⢱
⠀⠀⠀⠆⠋⠀⠀⠀⢀⠀⠀⠀⠀⠄⠈⢹⣷⣤⣧⣌⡋⣨⣀⡖⠀⠀⠀⠤⢀⡄⠀⠙⠋⠋⠀⠀⠀⠀⠙⣿⡧⠴
⠀⠀⠀⢧⠁⠀⠀⠀⠊⠀⡆⠀⠀⠀⢠⣿⣿⣤⣼⣿⣿⢷⣦⣤⣶⡀⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠠⢣⡄⠀⡆⢠⠚⢁⢱⣪⡶⠟⠻⠿⣿⣿⠟⡿⣷⣿⣿⣭⠟⠷⠀⠏⡠⡖⠒⢛⣄⢀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠖⠁⠀⠀⠀⠀⢸⡅⠀⠀⠀⠉⠝⠉⡄⢀⡀⠈⠱⠀⢀⣈⠿⠢⠛⠆⠠⠤⠖⡲⢄
⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠛⠓⠒⠛⠀⠘⠁⠀⣀⠄⠀⢁⠈⠡⠖⠀⢌⠉⠀⠀⠀⠀⠀⠀⠈⢤⡆⣸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣄⣀⠀⢀⣠⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⣾⠀⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠙⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡴⣛⣭⣭⣮⣭⣭⡓⢦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⢫⣾⣿⡿⠛⠈⠻⣿⣿⣦⡹⡎⢻⣿⡿⠿⣿⣿⠿⣿⣟⣯⣵⣿⣿⣿⠁⠙⣿⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⡅⠺⢸⣧⣼⣿⣿⣷⢹⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣧⣀⠈⠀⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡿⣷⢠⡉⢻⣿⣿⣿⢸⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⢷⡹⣿⣿⣧⣈⠈⣁⣼⣿⣿⢣⡟⣴⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠳⣬⡻⠿⣿⣾⡿⠿⣛⡵⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠛⠉⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠘⠃⠊⠁⠀⠀⠈⠀⠐⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣔⠀⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠓⠢⢤⡀⠀⠀⢠⡐⢧⡀⢠⠐⠀⢠⢦⠀⠀⡀⠀⢀⣤⣀⣀⣀⣘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⠀⠒⠶⣂⣀⡉⣶⡉⢨⡓⠄⠢⣤⣙⠚⠛⡉⠛⡦⡟⢋⡙⢻⠻⠛⢩⠍⠙⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⢀⠘⠿⠷⠶⠄⠹⣧⣌⣻⣦⡍⠁⢒⣻⡃⠈⢱⡆⣀⠐⠀⢀⠁⢈⠂⠀⠉⣠⠾⣍⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣁⣘⡋⠀⠀⠠⢤⣀⠀⠕⠂⠛⠻⠯⠂⠰⠿⠈⢐⠒⠞⠁⠀⠈⠁⠂⢀⠀⢠⣾⠙⢦⠈⠻⣦⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣛⠉⠉⠛⢡⣤⣀⣀⣰⣍⠓⠠⠀⠀⠀⠀⢀⠉⠀⠉⡁⠀⠀⠈⠠⠔⠀⡁⠀⢿⣍⠁⠀⠀⠀⠀⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣶⣦⣤⣤⡌⣙⠀⠀⣤⣄⠀⠀⠀⠀⢤⡾⠿⠏⠋⠉⠀⠀⠀⣠⠾⠾⢦⣽⣷⣄⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⡶⠞⠉⡙⠋⠉⢐⣠⠤⠈⠁⠦⠀⠉⠁⠀⠀⡸⠿⢷⣦⣄⡀⠀⣀⠀⠀⢡⣶⣶⣦⣽⡿⢿⣿⣦⣄⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣽⣿⡶⠛⣻⠛⠑⡾⠟⠓⢶⡀⠀⠀⠹⣿⣆⡙⢷⡈⠿⢶⣍⡻⣿⣿⣄⠀⠀⣿⣿⣿⣿⣿⣿⣦⣽⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣽⣟⢉⡩⠛⠲⠿⠷⠀⠴⣄⠙⠒⠠⠀⠙⢷⣄⠀⠀⢚⣲⣿⣗⡛⠻⣿⣷⣴⣿⣿⣿⣿⣿⣿⡿⢻⠕⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠱⠿⠷⠦⠶⡆⠘⠒⢂⠀⠀⠰⠂⠀⠀⠀⠀⠈⠀⠸⠿⢷⣮⣽⣦⡸⠏⢻⣿⣿⣿⠿⣿⠽⠊⠁⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠶⠛⠓⠲⢶⣶⣦⣦⣤⣵⣷⠎⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⠛⠿⣽⣷⣤⣹⣷⣾⣷⠄⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⢀⢀⣠⣶⣶⣶⡦⢤⣤⣌⡛⠏⣛⢛⠛⢿⡏⠘⡁⠀⣠⡀⠁⠀⠐⡦⠀⠀⠀⠀⠁⠙⠢⠄⠀⣾⣯⡉⣛⣿⣿⣧⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣈⣉⣉⣭⣿⣿⣿⣿⡽⠻⢷⡤⣤⡁⠑⠢⣈⢿⡁⠈⠉⠑⠒⠒⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠻⣿⡿⠇
⠀⠀⠀⠀⠀⠁⢉⣿⣿⠿⢿⣿⣿⡏⠻⠀⡙⡞⣾⡻⣶⡓⠈⣰⠏⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⡀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠠⠖⣶⣿⡿⢿⢓⠝⠋⠴⠁⠀⠀⠀⠀⠈⣋⠡⠽⣿⣦⠟⠂⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⢦⣀⡁
⠀⠀⢰⣿⣿⠋⠋⠐⠛⠠⠈⠀⠒⠀⠀⠀⠀⠀⠘⠃⠀⠁⢻⡆⠆⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠃
⠀⠀⠸⠟⡋⢠⠊⠀⠊⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠐⠤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠒⠆⠈⠀⠀⠈⢿⣦⡐
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠞⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⢭⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⡁⠙⠗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣻⡷⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⢦⣤⣤⡾⠷⣀⣈⣿⠂⠔⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠟⣔⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⣤⡄⠀⠀⠀⠀⠀⠈⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢸⡿⣾⢿⡝⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠈⠉⡴⠤⣤⠚⠀⠊⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⣍⣉⢺⡍⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠾⠁⠀⢹⡤⠀⠀⠈⠂⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⢉⢙⣷⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠻⣏⣩⣩⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣙⠗
⣀⣀⣀⢀⣀⡀⡀⣀⠀⣀⣀⢀⢀⣀⢀⣀⡀⢀⣀⠀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⠀⠀⢀⠀⣀⣀⡀⡀⢀⠀⣀⡀⢀⡀⠀⣀⡀⠀⠀⡀
⣿⠉⠉⣿⢛⠛⢛⢛⡛⢻⡛⣻⡛⠛⢻⠟⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣻⣿⣿⣛⣛⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣶⣶⣄
⠀⠀⠀⠀⢀⣠⣴⣶⣤⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡀
⠀⠀⠀⠿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣸⣿⡿⠿⠿⠿⠿⠿⢿⣿⡿⠛⠻⠟⠀⣤⣤⣤⣀⣤⣶⣿⣿⠉⠇
⠀⠀⠀⠊⠙⠋⢛⣛⠉⢙⣿⣦⠀⠀⠒⠀⠃⠉⠉⠉⠁⠐⠚⠓⠃⠀⠀⠀⢶⣒⡿⢙⠛⠷⠿⠋⠁
⠀⠀⠀⠁⠈⠉⠉⠉⠉⠋⠉⠛⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠄⢈⣟⠟⠀⠉⠈⢿⠆
⠀⠀⠀⠀⠀⢶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣦⣄⠀⠀⠀⠀⠀⢀⣨⣀⢂⠀⢀⡀⣨⣇
⠀⠀⢠⣴⣶⣾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⠿⣿⣟⠉⠀⠀⠀⠀⠀⠈⠉⠉⢉⣉⡀
⠀⠀⠸⠯⠼⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⣀⣀⣀⣀⣀⣄⡀⠀⠙⠂⢤⣷⣞⡿⠠
⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⠀⠀⣠⣶⣿⣿⡿⣉⣾⣿⡿⣿⠁⣼⠁⡀⠀⠸⣁⡼⠁⢧⣠⠇⠀⣠⡀
⠀⠀⠀⢀⣠⠎⢝⢽⣿⡿⠁⠀⢰⣿⣿⣿⣿⣿⣿⣶⣿⣷⢻⣦⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⣥
⠀⠀⠀⡿⢋⡴⢾⠖⠋⠁⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⡿⠈⠻⡇⠀⠀⠀⠀⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠃⠀⠃⠀⠀⠀⠀⠀⣼⡿⣿⡟⠀⠀⠀⠀⠀⣿⢿⠃⠀⠀⠱⣄⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠠⡀⠀⠀⡀⠀⠀⢀⣤⣼⣿⠿⠷⠤⠤⢤⣤⣤⡿⠺⠆⠀⠀⠀⠉⠀⠀⠀⣠⡼⠋⠋⠄⠄
⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠉
⠀⠀⠀⣿⣥⣴⣾⡿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠿⠿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⡀
⠀⢀⣠⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⢿⠛⢻⡿⣿⡆⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠰⠛⠉⠁⠘⠳⣿⣿⣿⣿⣿⠿⣿⣻⣿⠏⠀⠀⠀⠋⠘⠀⠼⠀⠈⠂⢸⠹⣿⠋⠙⠛⠛⢿⣏
⠀⠀⠀⠀⠀⠀⢠⣿⣹⣇⣀⣀⣀⣈⣿⣹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠆⡏⢀⣀⣀⣀⣘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣤⣶⡄
⠀⠀⠀⠀⠁⠀⣀⡀⣤⣀⠀⠀⠀⡠⢢⠄⠀⠀⠀⠀⠀⠀⣰⣿⢫⠖⣒⣍⠛⡩⢒⣲⡌⣿⣆⠀⠀⠀⠀⠀⠀⠀⣠⣄⣿⡏⣹⣿⡋⠅⣾⣦
⠀⠀⠀⠀⠀⢀⠒⠀⠰⠎⢵⠀⠀⡞⠛⠇⠀⠀⠀⠀⡴⣿⢿⣿⡸⣄⣿⡿⠀⢆⣸⣿⢇⣿⡿⢿⢦⡀⠀⠀⢠⣾⣿⣿⠉⠁⠙⠃⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⡀⠣⠤⠶⠞⠁⠀⠀⠀⠉⠀⠀⠀⠀⡎⠀⡿⠈⠻⣿⣶⣴⡾⠀⢷⣦⣴⣾⠿⠡⢸⡆⢳⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣴⣷⠉⠐⣠⠁⠁⠂⠀⠀⠈⢩⡄⠂⠨⣸⣧⣼⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠘⣯⢈⠡⠂⣤⠈⠃⣈⠘⡁⣤⠙⠘⠃⣼⠋⠏⠀⠀⢸⣿⡿⠀⣠⣼⣧⣶⣀⣀⣦⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢢⣾⡁⢴⠆⠄⣠⡀⢩⢀⣄⠠⠐⠇⢰⣇⡀⠀⠀⠀⠀⢻⡇⣾⣿⠙⠛⢹⣿⣷⡿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠐⠰⠆⢨⠁⣤⡀⠉⠰⠆⢃⣾⠏⠀⠀⠀⠀⠀⠀⠙⢿⣀⣶⡆⣠⣿⣿⠷⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣀⣀⡀⠀⠉⠀⢀⣀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⢿⣵⣾⣿⣯⣽⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⡿⠿⠗


⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⡀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢰⡧⢄⠀⠀⠀⡀⣾⡄
⠀⠀⠀⠸⢃⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠃⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡄⠀⡾⢿⣿⣿⣧⠀⠀⠀⠀⠀⣠⠄⠀⠀⠁⠀⡠⠀⠰⢤⡀
⠀⠀⠀⠀⣸⠋⢠⣶⡆⢸⡇⢰⣿⡄⢸⣿⡀⠀⠀⠀⠀⠀⢻⣿⣿⠇⣤⣄⣀⣤⡄⢿⣿⡏⠀⠀⠀⠀⣾⠁⣴⢛⢦⠀⠆⢀⠖⢲⡄⠙⣷
⠀⠀⠀⠀⢹⣆⠈⠛⢃⡜⢻⣌⡉⣁⣼⣿⠁⠀⠀⠀⠀⠀⠀⠻⢿⣦⡙⠹⡿⠙⣥⠿⠏⠀⠀⠀⠀⠀⢿⣄⠹⠦⠞⢀⣦⠘⠦⠴⠃⣰⠟
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣾⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠻⠶⠞⠋⠉⠙⠖⠶⠞⠋
⠀⠀⠀⠠⠀⢂⢠⣿⣿⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠁⠸⡟
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣌⢳
⠀⠀⠀⠀⢀⣠⠁⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠟⢃⠎⣤⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣽⢿⡷⣄⣀⢠⣴⣶⣦⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⢾⣆⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣿⠷⠦⢄⠁⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⡿⠇⠀⡿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠛

⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⣤⣶⣤⣤
⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⡩⠀⠀⢀⣀⣠⣤⣤⣤⢶⣤⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠐
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠚⠛⠛⠛⠯⠙⠋⠉⠙⣇⠀⠀⠀⡄⠀⠀⢠⣤⠀⠀⢠⡄⠀⢀⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠶⠛⠁⠀⠀⠀⢹⠀⠸⡇
⠀⠀⠀⠀⠘⢿⡇⠘⠻⠆⠹⣿⡈⠿⡄⠀⠀⠐⡄⠴⣾⣷⠆⠀⠀⠀⠀⠀⠁⠀⣀⠐⠂⠀⠉⠒⠀⠀⠀⠀⠲⠦⠖⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠋⠀⠘⣿⠁⠀⢀⣴⠻⡿⠉⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⢤⣤⡤
⠀⠀⠀⠀⠜⣿⣿⣿⣿⡿⠻⠟⠣⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠁⢀⠀⠀⢀⣠⠄⠀⣠⢦⠟⠉⠁⠛⠈⠁
⠀⠀⢠⡴⠃⠐⠇⠀⠹⠃⠀⠀⠀⠀⠀⠀⣤⣄⢰⡄⣀⢀⡀⡀⠁⠐⠀⠀⠠⠖⠩⠶⠃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠟⢿⠂⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⣰⢻⢠⠼⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢾⠅
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢠⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⠷⠶⢦⠟⠀⠀⠀⠀⣠⡴⠀⡿⢛⡒⠛⠛⢷⣦
⠀⠀⠀⠀⢸⣾⣿⣶⣶⣶⣾⣿⣿⠀⠀⢲⣶⣶⣶⡆⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠼⠵⠷⣦⣴⣿⣿⣷⣶⡸⣿⡇
⠀⠀⠀⠀⠘⣿⣿⠿⠿⣿⡿⠟⠛⠀⠀⠞⠉⠉⢹⠟⠐⠧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⡻⠉⠽⠟⠉⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣀⣤⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⠃⠀⠀⠀⠀⣀⣀⣀⣔
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣴⡷⠀⠀⡞⠉⠉⠙⠁⠀⠀⠀⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⣾⠃⠀⡾⢇
⠀⠀⠀⠀⠀⣶⣿⠷⡶⢿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⢺⡁⠉⢸⣿⣿⡏⠈⠙⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⡶⢿⠇⠸⠀⠀⠧⠀⠀⢹⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠉⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣤⣽⠃
⠀⠀⠀⠀⠀⣀⣀⣀⡀⣠⣤⣠⣶⣤⠀⠀⠀⠙⠀⠀⠀⠁⠀⠀⠠⠶⢾⠷⠶⠂⠀⠀⠀⠀⠰⣤⠖⠀⠀⠐⠛⠀⠀⢈⠁
⠀⠀⠀⠀⠸⣿⣿⣿⣶⣿⣿⣿⠿⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣲⣄⠠⢴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣿⡿⠛⠛⠛⣿⡏⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⠾⠃⠀⠀⢸⣿⡷⠿⠷⠿⣿⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⡴⡿⠛⠉⠛⣿⠀⠀⠀⠀⠁⠈⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠂⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠂

⠀⠀⢀⠠⠄⠂⠀⠀⠀⠐⠂⠀⠄⡀
⠀⣾⣿⣷⣠⣀⣠⢤⡀⠀⠀⠀⠀⠈⡑⣄
⠀⢻⣿⣿⣿⣧⣀⣤⣼⠁⠀⠀⠀⠀⠈⠺⠴⣄
⠀⢸⡿⣿⣿⣿⡟⢻⡿⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠒⠄⢀⡀
⠀⠈⠛⠇⠈⠉⢹⣿⡿⠂⠀⠀⢀⠂⠀⢀⠀⠀⠀⣄⡀⢦⡉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢋⠁⢠⡌⠁⠈⠠⠪⣦⠂⠀⠈⠙⠉⠉⠉⣴⠦⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢮⡙⡋⠃⣆⢀⣥⣿⣇⠀⠀⠀⠀⠀⣀⡄⡈⡄⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣶⣿⣿⣿⣿⣿⣿⠌⡁⠀⠀⢠⣦⢀⡀⠐⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣯⣿⡑⠅⡀⠀⠃⠀⠀⠔⠈⠀⠀⠀⠀⠱⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣽⣿⣿⣷⣄⠀⠦⠠⠄⢠⠠⠀⠀⠀⠀⢀⠀⠘⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⣿⠋⠳⢷⣶⣤⡀⠀⢠⠀⠀⣦⣬⠈⠀⠀⠘⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠱⠔⢊⣤⠤⢾⣿⣿⠿⠟⢲⣶⣿⣿⠲⣿⣶⣤⠁⣙⡀⢹⡙⠳⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠅⠊⠀⠀⠀⠀⠉⠑⠻⣿⣿⠀⠀⠉⠀⠀⠀⠙⢅⠳⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡑⠂⠐⠀⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠈⠉⠀⠰⠀⢀⠂⠀⢶⡔⠡⠐⡄⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠁⣀⠐⣄⠈⠲⣦⠀⠀⠆⠀⠀⠀⢀⠈⠀⡐⠀⠈⠂⠀⠀⠄⢮⣱
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠘⠂⠀⠈⠁⢀⣀⠀⠀⠄⠡⠀⠀⠀⠁⠀⠐⠀⠀⢀⠈⠱⡃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⣀⠑⢐⣀⣀⠠⠀⢠⠁⠁⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⣀⣁⣤⣼⡶⠂⠀⠀⠀⠈⢐⡄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠀⠄⠀⠈⠉⠘⠿⠋⠊⣀⠀⠘⠀⣄⠀⠁⠀⠀⡀⠀⠀⠀⢠⠐⠀⠤⠀⠀⠀⠀⢐⠀⠀⢀
⠀⠀⠀⠀⢐⠀⠀⠀⡄⠀⠀⠀⠀⠒⠃⠉⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠂⠀⠈⠘⡤⢔⡊⠁⠘⠠⠀⠀⠀⠂⠀⠀⠈⠠⡀⠘
⠠⠀⠌⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⢈⣦⠞⠡⠀⠉⢦⡀⠀⠈⠁⠀⠀⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢈⣠⠄⠀⠀⠀⢠⠈⣂⠀⠀⠀⠀⠀⠀⠀⣴⡟⠣⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠒⠂⣤⣀⣀⣀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠂⠀⠀⠁⠀⢀⣴⣾⡏⠁⠀⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⢀⠀⡄⠀⠤⢀⠠
⠀⠀⠀⠀⠈⠙⠛⣿⣿⡆⠀⠀⠀⠀⠐⠠⠄⠂⠀⠐⠒⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣰⣶⠶⣶⣶⣦⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠉⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠭⠴⠛⠉⠉⡩⢲⠎⢩⡏⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⣉⣀⡀⠀⠀⠀⠀⢛⠈⣢⣉⣴⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠺⠑⠒⠀⠈⠉⠒⠚⠳⠦⣀⡀⣠⣾⣿⣫⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡼⢟⠒⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡿⠧⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣒⠭⠒⠁⠀⠈⢄⡀⠀⠐⠭⠩⢛⣶⡦⣄⢀⢆⢆⣿⣿⣿⡿⠏⣏⣠⣷⣿
⠀⠀⠀⠀⠀⠀⣀⠠⠔⣊⣥⢖⠙⠓⡀⠀⠀⠀⠀⠀⠉⠒⠒⠚⢈⡡⣪⣾⣿⠀⢫⣾⣿⡿⠟⢁⣶⢿⣿⢻⢫⡇
⠀⠀⠀⠀⠀⢀⣿⢴⣢⣉⡁⢦⣀⠑⠒⡀⠀⠀⠀⠀⠀⢀⡠⣒⢵⣾⣾⣿⣿⠁⣿⣿⡟⠟⠚⠻⣿⣕⡕⣵⠋
⠀⠀⠀⠀⠀⠀⡏⠑⢶⡬⣉⡓⠶⣭⣕⣲⠊⣩⡽⠟⡩⢵⣺⣵⣿⣿⣿⣿⣯⣖⠛⠈⠀⠀⣀⡄⣿⣮⣾⠋⠱⠶⢤⣀
⠀⠀⠀⠀⠀⠠⡏⠀⠀⠙⡑⠪⢝⣒⢿⣾⣽⣓⣞⣿⣾⢿⣿⣿⣿⠿⠿⠏⠓⠒⣦⣤⣄⢦⡛⣄⣿⡿⣙⣑⣦⠀⠩⠃⣩⡕⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢭⣛⡲⠬⣅⡂⠉⢘⠛⣿⣶⣿⣧⣟⠛⠈⠤⠤⠔⡪⣻⣯⡟⢺⠁⣹⠟⠋⠉⡀⠺⢕⡶⢄⣠⣶⣷⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠳⠦⢌⣙⡋⣼⡿⠟⠋⠬⠍⠀⠀⠀⢀⢞⣵⢿⣿⡏⢃⡽⠓⠀⢀⠔⠁⠀⠀⠈⣉⡚⠻⢿⠷⢇⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⢉⠴⠉⢰⡁⠒⠂⢬⣙⣋⠉⠁⠀⠀⣀⣀⣀⢔⣕⣟⡷⣿⣿⣦⠯⠄⠀⠒⠁⠀⠀⠀⠀⠤⠀⠀⠀⡜⠀⢄⢨⠉⣱⡆
⠀⠀⠀⠀⠀⠀⢐⡯⠬⠥⠉⣀⣀⡒⠊⠭⠭⠶⣢⡤⠤⠊⢁⠖⣃⢴⡿⡯⣾⣾⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⠂⣸⢄⣠⠴⢊⠥⢻⠄
⠀⠀⠀⠀⠀⠀⠀⣿⠤⢂⣐⡒⠒⠀⠭⠭⠤⠐⢀⣀⣀⡨⡭⠍⠐⣫⣺⣿⣿⣏⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⡚⠭⢒⡩⣴⣫⣵⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢅⣒⣒⣒⡰⠯⠭⠭⠥⠔⢛⣒⣐⡤⠭⡍⡭⣿⣿⠿⡟⠙⠿⣿⢤⣘⠓⢬⠑⠲⣔⡢⠭⣐⣻⣵⣶⣿⢿⣿⠿⠛⡋⢬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢕⠬⡛⢌⡙⠩⢋⢟⡹⡋⢞⠑⡋⠛⠝⡒⢛⢷⣕⡢⠀⠀⠀⠹⢦⢽⣒⣺⣽⣕⣛⣭⣶⣿⡯⠟⢛⠩⢔⣊⣥⠔⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠉⠢⢤⣄⡀⢀⣁⣈⣀⡁⠈⣈⣰⣥⡿⣻⣄⠀⠀⠀⠈⠃⢰⣿⣿⠿⢟⢚⣉⣥⡒⠥⢒⠮⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠨⣛⢥⡀⠴⠚⣋⣽⢿⣿⣿⡿⣿⣟⠝⠒⢄⡀⠀⠐⣗⣖⣯⣽⠿⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠸⡩⡉⠹⢿⣻⣷⣟⣯⡿⠟⠊⠀⠀⠀⠀⠈⠁⠊⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⡀⠈⡆⢐⣿⡿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢐⠀⠂⠐⡠⠂⠔⡠⠂⡀⠀⠄⡠⢀⠀⠀⠀⠠⢀⠄⠀⠀⡠⢂⠐⠀⠀⠠⢀⠄⠀⢀⠔⡠⡠⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠊⠔⠥⢂⠔⢈⢔⡠⢊⠔⢊⠔⡠⢂⠑⡨⠐⠡⢊⠐⡡⢂⠀⡠⡠⢊⠄⠁⢒⡑⡠⢂⡔⢃⠔⡡⢪⠊⠔⠑⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠘⠁⠊⠁⠊⠐⠁⠊⠐⠡⠂⡠⢊⠔⠡⠪⠊⠀⠁⠊⠀⠋⠀⠁⠈⠀⠁

⠀⠀⣀⠤⠶⣤⣄⣀⣤⡴⠤⢀⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⢀⡶⠛⣉⡛⠟⢋⡙⢳⣆⠀⠀⠀⠀⢸⣷⠟⠛⠶⣶⠞⠛⢧⣿⠀⠀⠀⠀⢠⡟⢉⣬⡙⢋⣥⡉⢻⡀
⠀⠀⢸⣇⠸⣿⠗⡸⢿⡿⢠⣿⠀⠀⠀⣀⣸⣿⡀⠺⢷⣇⠀⠿⣾⣟⡀⡀⠀⠀⣸⣇⡘⠛⣣⡜⠛⢁⣼⣇
⠀⢠⡏⠙⠓⠒⠚⠟⠒⠒⠛⠙⡆⠀⠀⢹⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⡟⠁⠀⢠⣿⣿⠉⠉⠈⠁⠉⢩⣿⣿
⠀⠸⣇⠀⣠⣴⣶⣶⣦⣤⡀⢠⡇⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠈⢿⣿⠃⠀⠀⠀⠀⢸⣿⡿
⠀⠀⢻⠖⢿⣿⣿⣿⣿⣿⠗⢿⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠻⢄⣀⣀⣀⣀⡤⠟
⠀⠀⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⣉⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠿⠟
⠀⠀⡠⣰⢀⣠⣤⣤⣤⣀⢰⡤⡀⠀⠀⠀⣀⣠⣄⣀⣈⣋⣀⣀⣤⣀⠀⠀⠀⠀⠠⣴⣶⣦⣤⣤⣴⣶⣦⠄
⠀⠀⢉⡭⠟⠯⣝⢟⡭⠟⠿⣏⠁⠀⠀⠀⣸⠿⠛⣉⡛⠟⢉⠙⠷⢧⠀⠀⠀⠀⠐⠑⢁⣀⡙⢋⣀⡈⠊⠂
⠀⠀⢾⠀⠠⣶⢸⣿⢠⡶⠀⢸⠀⠀⠀⠀⢿⣆⠻⣿⠇⠀⠉⠁⣰⣿⠀⠀⠀⠀⢠⡀⠀⠀⢠⣄⠀⠀⢀⡄
⠀⠀⢈⡲⠤⠤⠺⠿⠢⠤⠴⣊⠀⠀⠀⢠⢪⣿⣿⣶⣿⣾⣶⣾⣿⡟⢆⠀⠀⢠⡏⠀⠁⠈⠀⠀⠁⠈⠀⢸⡆
⠀⢠⣿⡿⠀⠠⠤⠤⠤⠀⠸⣿⣧⠀⠀⡏⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠉⠉⠈⢷⣄⠁⠃⢁⣴⠏⠈⠉⠁⠀⠘⠞⠙⢿⣿⣿⣿⣿⡿⠋⠙⠞⠀⠀⢹⠟⠃⢤⠀⠀⠀⠀⡤⠘⠻⡟
⠀⠀⠀⠀⠀⠙⠻⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠾⠶⠔⠢⠶⠷
⠀⠀⢸⣶⣤⣤⡀⢀⣤⣴⣾⡇⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⣄⣀⣀⣠⣤⣤⡀
⠀⠀⢸⡽⢋⡍⠻⠟⣩⠛⣏⡇⠀⠀⠀⠀⢨⠻⠋⣉⠙⠟⣉⡉⢻⢏⠀⠀⠀⠀⢠⢟⠋⣉⡙⠟⢉⡉⢻⢧
⠀⠀⢸⠣⡙⠿⣢⣌⠻⢟⠜⡇⠀⠀⠀⠀⢸⡀⠘⠿⠖⠈⠿⠗⠀⡼⠀⠀⠀⠀⠸⡀⠘⠿⠇⠀⠿⠟⠀⡸
⠀⠀⣾⣿⣆⣉⣙⣋⣁⣠⣿⣷⠀⠀⠀⢰⠛⢿⣶⣶⣷⣾⣶⣶⡾⠛⣦⠀⠀⠀⣴⣿⣶⣶⣾⣾⣶⣶⣾⣿⣆
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠸⡄⢸⣿⣿⣿⣿⣿⣿⡇⠀⡿⠀⠀⢸⣿⣿⠟⠋⠉⠉⠉⠙⢿⣿⣿
⠀⠀⠀⠈⠛⠿⣛⣛⠿⠛⠁⠀⠀⠀⠀⠀⠙⠾⢿⣿⣿⣿⣿⣿⠿⠞⠁⠀⠀⠈⠋⠁⠠⣀⣀⣀⣀⡠⠀⠈⠋
⠀⠀⠀⠀⠀⠑⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠯⠽⠗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢛⣦
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣿⣿⡇⠀⠚⣿⠀⠀⢀⣤⡾⠛⡆
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡧⠂⠀⣿⣦⣤⡞⠝⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⢀⠀⠐⠀⠀⠀⠀⣀⣠⣾⣿⣟⡻⣿⣿⣷⠀⣿⡷⠀⣼⣿⣿⣿⡇⠀⣀⡜
⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠈⠆⠀⣀⣀⣤⣤⠀⢈⣼⣿⣟⠋⠹⣿⣷⣜⣿⠟⢰⣿⣷⣾⣿⣿⣿⣿⣢⡼⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡠⠄⠀⠟⢦⣤⣶⣾⣿⣿⣿⣿⣾⣿⣿⣏⠙⢷⣦⣈⢿⡏⡟⠘⠺⣿⢿⢟⡿⣱⠈⡿⣿⣤⡀
⠀⢀⠐⠀⠐⢆⠀⢸⡏⠉⣀⢀⠈⡵⢜⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣷⣄⣿⡟⢼⡅⣿⢷⡸⠃⡁⣾⡇⢻⡀⣧⢹⣿⡿⣖⢄⡀
⠀⠀⠀⠀⡀⠀⡆⠈⠅⢠⠀⠰⡆⠀⣁⠋⠸⢻⢿⣿⣿⣿⣿⣷⡄⢙⣿⣏⠼⣷⠸⣿⡸⡎⢣⢣⣧⢹⡿⣏⠷⠹⢸⡇⠻⣿⢟⢮⣀
⢀⡀⠀⠀⢴⠎⠉⠀⠦⠀⠀⠀⠀⠀⠁⠀⠐⠁⠈⠁⠙⠻⢿⣿⣿⡏⢹⣿⠀⣿⣧⢹⡇⢇⢸⢻⡙⢷⡁⢩⣶⣶⣿⣮⠃⣿⣷⡁⢻⡄
⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⢠⣤⣀⣤⣀⠀⢀⠀⠀⠝⠀⣰⡟⢿⣇⠈⢿⣧⠸⣿⡘⡇⢸⢸⢸⣇⢸⣇⣤⡻⠿⠟⠙⢷⡮⡛⢄⢀⠹⡄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣿⣿⣿⣶⡿⢁⠀⠀⢰⣿⡄⢸⣿⣆⠀⢻⣇⢻⡇⣿⢸⢸⢸⣿⢸⣿⢹⡛⠷⠾⣿⣶⣍⠪⡪⡓⠡⡙⣆
⠀⠀⠀⠁⠀⠀⠀⠀⣀⣾⣿⣿⠟⠛⠛⣿⣿⣿⡁⠀⠐⠉⠙⢛⡌⣯⢿⣧⠀⢻⡞⣧⢸⣸⠀⢸⠇⣾⣿⡄⠻⢶⣤⣤⣄⣈⡙⢮⣎⠀⠙⢮⣧⡀
⠀⠀⠠⡔⠛⠊⠀⠉⠉⣿⣿⣿⡀⠀⠀⣼⣿⡟⡷⠆⠐⠀⣀⢹⣧⠈⢎⢻⣧⠠⣿⣿⣬⣿⠀⣿⢠⡇⣿⡇⠀⣠⣽⣶⠿⠟⠛⠛⠿⣷⣾⣷⣶⣿⣄
⠀⡄⠈⡁⡀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣟⠢⠤⠠⡄⠀⡼⣿⣷⡀⠣⡻⣧⠻⣿⡹⣿⡀⣿⠟⠀⣿⡇⠀⣿⡟⠀⣠⣖⣚⣿⣿⣾⣿⣿⣿⣿⣟⣆
⠰⠂⠁⠄⠀⠀⠀⠀⠀⠘⠓⢹⢿⢿⣿⣿⡿⣍⠁⠀⠀⣀⣤⢷⣿⣿⣷⡀⠹⣿⣆⠹⣷⡘⢷⣿⢸⣧⣿⡇⠀⢻⡄⢸⡿⠿⠉⠛⢿⣿⣿⣿⣿⣿⣿⣜⣆
⠀⠀⠀⠀⢘⠳⠆⡀⢰⠃⠀⠀⠊⢘⠀⠀⠀⠈⠆⠢⠀⣈⡟⠀⢸⣿⣿⡷⡀⢻⣿⣆⠈⢷⣌⣿⡟⡿⠿⢿⡀⣆⢷⠸⡇⣴⣾⣿⣛⣻⣿⣿⣿⣯⣽⣽⣿
⠤⠀⣀⠄⠈⠂⠋⡄⠈⠠⡂⠀⢀⡄⠀⠀⠀⠆⢰⣄⡤⠉⠀⠀⠈⢻⡟⣿⣤⠤⠛⠻⣆⠀⠻⣿⡟⣠⠀⠀⠈⠙⠊⠧⠷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠃⡀⠀⠁⢀⣮⠀⠀⠆⠃⠀⠈⠆⣀⠀⠀⣴⠤⡀⠀⠀⣶⠟⢻⠄⠽⣿⠁⢀⠀⠠⣶⡆⠲⠉⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠉⠁⠀⢀⡂⢃⡀⠈⠀⠁⢈⡅⠙⡄⠀⠒⠀⠀⠶⢿⣧⡀⠽⣧⣼⡀⢀⠤⠀⠀⠀⣠⣍⢧⣤⣤⢄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉
⠀⠀⠁⠀⠀⠀⣀⠀⠉⠙⠺⠋⢂⠀⠀⠊⠀⠀⠈⠐⠂⠀⢀⣀⠘⠋⠃⠀⢳⠄⠹⢅⠀⠶⡃⠀⠿⠁⠀⣍⠷⣿⢦⡠⠁
⠀⠀⠀⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣟⠂⠀⢀⠀⠘⠠⠀⠈⠑⠄⣠⠀⡆⠀⠀⠘⠢⠈⣿⡗⠀⠀⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠳⣧⡀⠀⠈⠉⡓⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡆⠈⠻⢟⣉⣁⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣄⠀⠈⠉⠑
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣦⣄⠀⣠⣤⠶⠶⠛⠛⠛⠛⠛⠉⠛⠛⠛⠻⠶⣶⣤⠀⣀⣤⠶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏⠻⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⡤⠤⠦⣄⡀⠀⠈⢻⡄⠀⠀⠀⣾⠃⠀⢀⣤⠶⠦⣤⡀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡏⣠⣶⣶⡌⢷⡀⠀⠀⣿⠀⠀⢸⡇⠀⢠⡟⢠⣶⣶⡄⢻⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣈⣷⠀⠀⠀⠸⣇⠻⢿⠿⢃⡾⠁⠀⠀⣿⢀⡀⢸⡇⠀⠘⣇⠸⢿⡿⠃⣼⠃⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢠⡟⠘⣧⡀⠀⠀⠈⠓⠶⠖⠋⠁⠀⠀⣸⢧⠾⣷⡈⣧⠀⠀⠈⠛⠶⠶⠛⠁⠀⠀⢀⣼⠛⣧
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠘⣮⡿⠃⠘⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢿
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠶⠛⠋⠀⠀⠀⠘⠃⠀⠀⠈⠙⠳⠶⠶⠴⠶⠚⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠘⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿
⠀⠀⠀⠀⠀⣷⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡏
⠀⠀⠀⠀⠀⢹⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣸⠇
⠀⠀⠀⠀⠀⠘⣧⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⡿
⠀⠀⠀⠀⠀⠀⢿⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⡇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢰⡇⠙⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢻⡇⢸⡆⣼⠃
⠀⠀⠀⠀⠀⣀⣀⣀⣈⣳⣏⣀⣀⣇⣈⣙⣓⣦⣤⣀⣀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣟⣁⣀⣼⣁⣸⣷⣃⣀⣀⣀⡄
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠷⠯⠤⠭⠽⣟⡿⠭⠯⢿⡽⠧⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣀⠀⠀⠀⠀⠀⣀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠿⠿⠟⠻⠿⠿⠿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⢁⣀⣤⠤⠶⠶⠶⠶⠤⣤⣀⡈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿
⠀⠀⠀⠀⠀⠀⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡈⠉⠛⠒⠶⠶⠶⠶⠒⠛⠉⢁⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣶⣶⣦⣤⣤⣶⣶⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣻⣿⣛⣻⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠴⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠉⢁⡠⠂⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⣨⠏⠀⠈⠁⢀⡄⠊⣀⠴⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠄⠐⠀⡈⠀⠀⠐⡋⠁⠀⠀⠀⠂⣁⡤⠚⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠤⠠⣤⠖⠉⠉⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⠄⠀⠀⠀⠀⡀⠄⢀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠢
⠀⠀⠀⠀⢘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠪⣷⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤⣄⠀⠀⠀⢀⡀
⠀⠘⠷⣄⠂⠈⢽⣇⡕⠏⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠳⠲⠄
⠀⢠⣄⠙⢻⡀⠱⢃⣴⣤⣧⣦⣶⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀⣀⡀
⠀⠀⠨⢧⠈⢥⣠⢀⣌⡀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠑⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠈⣣⠜⣉⡁⠙⠛⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡜⢲⠟⠛⠷⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡀⡆⠃⢀⢠⠇⡄⢠⠃⡄⢀⠄⠀⠀⠀⠀⠂⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡰⣌⢆⠣⢱⣠⣘⣘⡰⠡⢃⠎⡰⢋⠀⠀⣀⠀⠀⠀⢀⠀⡀⠀⢀
⠀⠀⠀⠠⠔⠀⢀⠀⠈⣇⠀⠀⠀⠐⠦⣄⡐⠦⣀⠦⡑⢌⠾⠋⠉⠀⠀⠀⠉⠓⢮⢊⠴⣋⠴⠊⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡀⠀⢈⠳⠀⡿⠒⠒⠒⠠⠤⣀⣉⠓⠬⡑⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠵⢚⣁
⠀⠀⠀⠀⠁⠀⠈⠂⢧⢈⡉⠓⠶⠆⠀⠠⠬⢭⡑⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠭⠤⠒⠒⠉⠁
⠀⠀⠀⠛⠀⠀⠀⠨⠶⢦⡉⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈
⠀⠀⠐⠂⠀⠈

⠀⠀⡀⡀
⣤⣤⣉⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣿⢻⡟⣿⣿⢻⡟⣿⣿⢻⡟⣿⢻⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣼⣧⣿⣿⣼⣧⣿⣿⣼⣧⣿⣼⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⢻⡟⡟⣿⢹⡟⡟⣿⢻⡟⡟⣿⢻⡟
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣷⣿⣾⣷⣷⣿⣾⣷⣷⣿⣾⣷⣷⣿⣾⠷
⠀⠀⢸⢹⡏⣿⢹⢹⡏⣿⣽⡿⢟⣛⣭⣭⣭⣭⣛⡻⢶⣼⣏⡏⣿⢹⡏⡏⣿⢹⡏⣿⣿⢹⡏⣿⣿⢹⡇
⠀⣀⣸⣾⣷⣿⣾⣾⣿⢟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡻⣿⣿⣾⣷⣿⣿⣾⣷⣿⣿⣾⣷⣿⣿⣾⣅
⠀⣿⢹⣿⡏⣿⢹⡿⣡⣿⣿⣿⣿⠿⠛⠀⠀⠛⠻⢿⣿⣿⣷⡌⢿⣿⢹⡿⡏⣿⢸⣿⡏⣿⢹⡏⡏⣿⢸
⠀⠙⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⡏⠀⠀⢰⣦⣤⣄⣾⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛
⠀⠀⣿⣸⢸⣿⡏⣾⣿⣿⣿⣿⣷⣄⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣧⢹⡇⣿⣸⢸⡇⣿⣸⢸⡇⣿⢸
⣤⣤⢿⣿⣿⣿⣇⢿⣿⣿⣿⣿⡿⢿⣿⣶⣦⠀⠀⠘⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⢿⣿⣿⣿⢿⡿⣿⣤
⣇⣿⣸⣇⣇⣿⣿⡜⣿⣿⣿⣿⣅⡀⠀⠉⠁⠀⣀⣼⣿⣿⣿⣿⢣⣿⣸⣇⣇⣿⣸⣇⣇⣿⣸⣇⣿⣿
⠉⡿⣿⢿⣿⡿⣿⢷⡜⢿⣿⣿⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⡿⣣⣿⣿⢿⡿⣿⣿⢿⡿⣿⢿⢿⡿⡏⠉
⠀⣇⣿⣸⣿⣇⣿⣸⣿⣦⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣾⣟⣿⣿⣸⣇⣿⣿⣸⣇⣿⣼⣸⣇⡇
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⠶⣮⣭⣭⣭⣭⣵⠶⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠀⠠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠈⢦⠁⢈⣀⠀⠄⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠁⠈⠀⠀⠀⠂⢨⠀⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣬⠐⠀⠐⠀⠠⣢⢠⢁⠀⠀⠢⢠⡾⠀⠉⠠⠿⠇⠄⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠾⢮⡁⠁⠈⠉⠹⣘⠉⣀⡄⠀⠀⠆⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠐⠁⠐⠋⠓⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣠⡇⢀⠶⣾⣤⣤⡿⡦⠀⡀⣾⣷⣴
⠀⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠉⠉⠀⠀⠁⠘⠚⠀⠀⠀⠀⠙⠀⠈⠁⠈⠋
⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠂
⠀⠀⠀⠂
⣀⣀⣀⣐⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣐⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢋⠁⡇⠦⢄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠯⣙⠶⠯⠝⠛⠋⠩⠥⠟⡦⢄⡀⠀⠀⠀⠀⠀⣠⡾
⠀⢀⡀⠀⢤⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⢔⠭⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⠓⢄⠀⢀⣴⡟⣡⠇
⠀⠀⠹⣦⣀⠹⣦⡀⠘⣷⡀⠰⡆⠀⣀⠔⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣏⣴⡟⠀⢀⡀
⠀⠀⠀⠀⠻⣷⡌⢻⣦⠸⣿⡄⣿⣀⢿⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣟⣿⣿⠏⠀⣠⡞
⠀⠀⠀⠀⠀⠘⣿⣦⡹⣷⣾⣿⣾⣧⢿⡇⢸⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠻⣿⣿⣿⣿⣟⣱⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣿⡝⣯⢿⣿⣿⣸⣷⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣝⢻⣿⣿⣿⣿⣿⣫⣦⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⢼⣿⣿⣿⣿⣿⣯⣽⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡹⣷⢻⣿⣿⣿⣿⢿⣿⣤⣴⣶⣄⡀⠀⠀⠀⡐⠀⣾⣿⡿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⡿⢿⠹⠞⠞⣿⣿⣻⣿⣿⡿⣦⡀⠐⠃⠀⣿⣿⡛⢿⣿⣿⣻⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣀⠤⠤⠖⠢⣀⠙⠉⢋⣟⢿⣿⣧⠅⠀⣰⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣩⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠉⠲⢄⠈⢦⢿⣿⠂⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣩⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠢⡀⠀⠀⠀⠀⠀⠀⠘⢆⢨⢉⠿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠆⠈⠲⢤⡀⠀⠀⠁⠀⢸⣷⣺⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⠀⠀⠀⠈⠢⠀⠀⣸⠟⠛⠉⠀⠀⣿⣿⣿⣿⣻⣿⣿⣿⡟⠁⠀⠀⠀⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡄⠀⠀⢀⡠⠔⠈⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⣿⣿⣿⣿⡀⠀⠀⠀⢠⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⠈⠀⠈⢀⠀⠀⠀⠀⠀⢀⢤⣜⠿⣿⣿⣿⣿⣿⣯⡁⠈⠀⠀⠀⢆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⢶⠏⠱⢆⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣛⣃⣼⠻⣿⣦⡐⣦⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⣦⡘⠙⠃⡀⢀⣈⣠⣤⢀⣤⡤⠤⡀⠈⠋⠉⢨⣍⡀⢹⣿⡟⣰⣏⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡄⢴⣾⡿⠆⠲⣟⠉⡉⠉⠛⠻⣿⣳⢰⣦⣴⣦⣚⣿⣿⣿⠄⣩⣾⣿⣿⣇⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠑⢮⣄⢠⣴⣾⣿⣿⣾⣿⣷⣽⠹⣩⣵⣿⣿⣿⣿⣿⡻⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣯⣟⣸⠦⠤⠶⠚⠿⠎⢻⣿⣿⣿⣿⣿⣁⠱⠀⡄⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⠏⢻⣿⣷⠏⡀⢢⢠⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⢹⣇⠘⠀⠈⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⠏⠀⠁⠀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠇⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢠⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢹⡀⠀⠀⠀⠀⢠⣋⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣙⡇⠀⠀⠀⢀⡏⠡⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠀⡇⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣨⣭⣷⠤⠤⠤⠼⠓⠶⠶⠗⢤⣀⣀⣀⣀⡀⠀⠀⠀⡄⢠⢸⢠⡟⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠿⠎⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣯⡭⠟⠃⠀⠀⠀⠸⠀⣮⡿⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⢷⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢁⡞⢠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡄⠀⠀⠀⠀⠀⠀⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⢣⣿⡑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡎⠃⠀⠀⠀⠀⠀⠀⠁⢠⡏⠀⠀⠀⢀⠀⢤⠤⠀⠶⠒⠒⠒⠀⠔⠒⠚⠛⠋⠁⠙⡆⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⡏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⣁⠀⠀⠠⠒⣂⡀⠀⠀⠊⠒⠆⠀⢀⠙⣆⢸⡓⠿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡀⠃⠀⡸⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠐⠉⠔⠒⠂⣀⣈⠉⢁⡀⠈⠙⠃⠀⢤⡀⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⢠⢰⡇⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡠⢤⣄⣊⠁⠉⠀⣉⣩⣥⡀⠐⢦⠈⠁⠀⡀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⣿⣸⢀⡎⠀⠀⢹⠀⣀⠀⠀⢀⡄⣸⡃⠀⠒⠲⣆⣀⣋⣡⠴⠖⠛⠉⠀⢂⠙⣦⠈⠀⠐⣄⠛⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣸⣿⡼⠀⠀⠀⢸⡇⠹⡆⠀⢸⣷⠏⠙⢦⠰⢾⠉⠩⡉⣧⠀⠀⠀⠀⠀⠸⣆⢻⡙⠒⠒⠓⢦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠛⠒⠛⠛⠉⠁⠀⠀⠈⣇⢸⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⠈⠳⣼⠆⠀⠀⠈⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠀⠀⠷⠏
⠀⠀⠀⠀⠀⠠⠀⠀⡀⡰⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠻⠋
⠀⠀⠀⠀⠀⠀⠡⡐⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠰⢠⠃⣰⠋
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣼⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡇⠀⣨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣿⠌⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠠⡤⠄⠀⠀⠀⠘⣧⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⡟⠀⠀⠀⠀⢸⡁⠀⢀⠀⡖⣦⢠⣹⣄⠀⣇⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⠳⣤⡼⠃⠓⠋⠀⠉⠈⠀⠈⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⣭⣭⠿⠀⠀⠀⢠⠈⠡⠤⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠠⢜⠀⠀⠶⠖⣿⠍⠡⠤⠤⢵⠀⢚⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣉⣿⣉⠘⣛⣛⣽⡇⠀⢒⣚⡿⠀⠥⠵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢤⣷⠀⢈⣁⣀⢽⡇⠈⣉⣭⣿⡀⢒⣲⡧⠀⠀⡀
⠀⠀⠀⠀⢀⠀⡀⡤⢄⡐⢒⡲⠒⠚⠓⠒⠛⠂⠀⠂⠤⢾⡃⠀⠶⣺⣯⠄⣉⡥⣿⢉⣽⡇
⠀⠀⠀⠀⠘⠙⠓⠻⢶⠀⠈⠀⠀⠀⠀⣀⣠⠴⠖⠒⠈⢉⠉⠈⢻⣽⡻⡤⠤⣶⣷⢴⣿⣧⡆⢀⣤⣤
⠀⠀⠀⠀⢀⡀⠀⢈⢹⠀⠀⠀⠀⠀⠀⡿⠿⠷⢤⣬⣦⡶⣰⣭⣿⣶⣒⣖⣒⣲⣾⣽⠿⠿⣯⣽⣿⣿⠷⠂
⠀⠀⠀⠀⠈⠙⠊⠛⠻⢦⣀⠀⠀⠀⠀⣏⠂⠀⠀⠀⠀⡏⠁⠉⠈⠉⠩⣏⠉⠉⠾⣁⣐⣐⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠤⠹⠷⢶⣦⣄⣀⣇⣠⣤⣶⣾⠽⠏⠂⠐⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⠿⢯⣿⣟⣿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡍⢊⢘⡞⣸⢫⡾⣻⡷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠫⠿⢠⠘⣾⡼⢡⠟⣵⢏⡿⢫⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣜⡅⢋⢍⠞⠀⢼⢠⡏⡼⢣⡟⣱⠟⣵⣿⡀
⠀⠀⠀⠀⠀⢀⣿⣌⣩⣾⡧⠷⠚⠻⡞⣰⢣⠟⣸⠇⡼⢃⣴⢟⡴⢠⢄⡄⣀⡀⣀⢀⢀⣀⣀⣤⣴⡶⣶⡶⢤⣄⣀
⠀⠀⠀⠀⠀⠈⠧⠽⠿⠋⠀⠀⠀⠀⠘⢧⡶⢛⣉⣼⢣⣾⠃⣾⢣⡟⣸⢣⡇⣿⣿⢨⡟⣫⡴⣫⡵⢛⣵⣾⠿⠛⠛⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⣫⣵⡾⢁⣼⢣⢟⣴⢫⣿⢃⡿⣾⣾⢰⡟⣼⠋⣴⡿⢉⣤⣶⡿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣡⠴⢛⣵⡿⢛⡅⣾⡏⣸⢧⣧⣇⣿⡟⣡⡾⢋⣴⠟⢋⣥⣤⣶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢡⣾⣫⣄⣄⠿⣘⣿⠁⡿⢸⣿⢸⡏⢰⡟⢠⡿⣡⡾⠛⣋⣁⣤⣴⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣾⢉⡭⢬⣓⣜⣿⠀⣿⢸⡇⣿⠀⣿⢀⡟⣰⢋⠔⣋⣩⣭⣥⡎⠀⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣿⡞⢛⣺⣽⣸⢀⡏⢸⡇⣿⠸⡇⢸⠀⡏⠿⣋⠵⠶⢶⢋⣿⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠿⢷⣒⠋⣩⠉⠙⠃⠛⠀⠛⠀⠓⠈⠁⠀⠀⠈⠭⢛⣭⠔⣲⡀⢹⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⢤⠎⠩⠭⢭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡟⠶⣚⣷⠄⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣚⠻⠀⠘⣿⡭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠇⠀⠑⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣡⠂⠀⠀⢨⠽⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡱⠆⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡏⠀⠀⠀⠀⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡤⠁⠀⠀⠀⢠⣇⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⠄⠀⠀⠀⢀⣻⡗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡼⠃⠀⠀⠀⣠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠰⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿